9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Trigonometric Ratios and Functions Right Triangle Trigonometry Angles and Radian Measure Trigonometric Functions of Any Angle Graphing Sine and Cosine Functions Graphing Other Trigonometric Functions Modeling with Trigonometric Functions Using Trigonometric Identities Using Sum and Difference Formulas Sundial (p. 518) Tuning Fork (p. (p 510) Ferris Wheel (p. (p 494) SEE the Big Idea Terminator (p. 476) Parasail iliing ((p. p. 46 5)) Parasailing 465) hsnb_alg2_pe_09op.indd 458 2/5/15 1:46 PM Maintaining Mathematical Proficiency Absolute Value 2 Example 1 Order the expressions by value from least to greatest: ∣ 6 ∣, ∣ −3 ∣, —, ∣ 10 − 6 ∣ ∣ −4 ∣ ∣6∣ = 6 2 ∣ −3 ∣ = 3 1 2 2 4 ∣ 10 − 6 ∣ = ∣ 4 ∣ = 4 —=—=— ∣ −4 ∣ The absolute value of a negative number is positive. 2 So, the order is —, ∣ −3 ∣, ∣ 10 − 6 ∣, and ∣ 6 ∣. ∣ −4 ∣ Order the expressions by value from least to greatest. 1. ∣ 4 ∣, ∣ 2 − 9 ∣, ∣ 6 + 4 ∣, −∣ 7 ∣ 3. ∣ −83 ∣ −5 ∣ 2. ∣ 9 − 3 ∣, ∣ 0 ∣, ∣ −4 ∣, — ⋅ ∣, ∣ −2 8 ∣, ∣ 9 − 1 ∣, ∣ 9 ∣ + ∣ −2 ∣ − ∣ 1 ∣ 4. ∣2∣ ∣ −4 + 20 ∣, −∣ ∣, ∣ 5 ∣ − ∣ 3 ⋅ 2 ∣, ∣ −15 ∣ 42 Pythagorean Theorem Example 2 Find the missing side length of the triangle. 10 cm 26 cm b a2 + b2 = c2 Write the Pythagorean Theorem. 102 + b2 = 262 Substitute 10 for a and 26 for c. 100 + Evaluate powers. b2 = 676 b2 = 576 Subtract 100 from each side. b = 24 Take positive square root of each side. So, the length is 24 centimeters. Find the missing side length of the triangle. 5. 6. c 12 m 7. b 9.6 mm 7 ft 25 ft c 5m 9. 8. a 10. 35 km 12 1 in. 3 7.2 mm 3 yd 10 b a 1 yd 2 21 km 4 in. 11. ABSTRACT REASONING The line segments connecting the points (x1, y1), (x2, y1), and (x2, y2) form a triangle. Is the triangle a right triangle? Justify your answer. Dynamic Solutions available at BigIdeasMath.com hsnb_alg2_pe_09op.indd 459 459 45 2/5/15 1:46 PM Mathematical Practices Mathematically proficient students reason quantitatively by creating valid representations of problems. Reasoning Abstractly and Quantitatively Core Concept The Unit Circle y The unit circle is a circle in the coordinate plane. Its center is at the origin, and it has a radius of 1 unit. The equation of the unit circle is x2 + y2 = 1. (0, 1) (x, y) θ (−1, 0) Equation of unit circle (1, 0) x (0, 0) As the point (x, y) starts at (1, 0) and moves counterclockwise around the unit circle, the angle θ (the Greek letter theta) moves from 0° through 360°. (0, −1) Finding Coordinates of a Point on the Unit Circle Find the exact coordinates of the point (x, y) on the unit circle. y SOLUTION (0, 1) (x, y) Because θ = 45°, (x, y) lies on the line y = x. x2 + y2 = 1 Write equation of unit circle. x2 + x2 = 1 Substitute x for y. 2x2 = 1 (−1, 0) 45° x (0, 0) Add like terms. 1 x2 = — 2 1 x=— — √2 (1, 0) (0, −1) Divide each side by 2. Take positive square root of each side. ( 1 1 ) ( — — √2 √2 2 2 ) The coordinates of (x, y) are — —, — — , or —, — . √2 √2 Monitoring Progress Find the exact coordinates of the point (x, y) on the unit circle. 1. 2. y 3. y (0, 1) y (0, 1) (0, 1) (x, y) 135° (−1, 0) (1, 0) (−1, 0) (1, 0) x (0, 0) (0, 0) (−1, 0) 225° (1, 0) (0, 0) x x 315° (x, y) (0, −1) 460 Chapter 9 hsnb_alg2_pe_09op.indd 460 (0, −1) (x, y) (0, −1) Trigonometric Ratios and Functions 2/5/15 1:46 PM Mathematical Practices Mathematically proficient students reason quantitatively by creating a coherent representation of the problem at hand. (MP2) Reasoning Abstractly and Quantitatively Core Concept The Unit Circle y The unit circle is a circle in the coordinate plane. Its center is at the origin, and it has a radius of 1 unit. The equation of the unit circle is x2 + y2 = 1. (0, 1) (x, y) θ (−1, 0) Equation of unit circle (1, 0) x (0, 0) As the point (x, y) starts at (1, 0) and moves counterclockwise around the unit circle, the angle θ (the Greek letter theta) moves from 0° through 360°. (0, −1) Finding Coordinates of a Point on the Unit Circle Find the exact coordinates of the point (x, y) on the unit circle. SOLUTION Because θ = 45°, (x, y) lies on the line y = x. y x2 + y2 = 1 Write equation of unit circle. x2 + x2 = 1 Substitute x for y. 2x2 = 1 (0, 1) (x, y) (−1, 0) Add like terms. 45° (1, 0) x (0, 0) 1 x2 = — 2 1 x=— — √2 Divide each side by 2. Take positive square root of each side. ( ) ( — — (0, −1) ) √2 √2 The coordinates of (x, y) are — —, — — , or —, — . 2 2 √2 √2 1 1 Monitoring Progress Find the exact coordinates of the point (x, y) on the unit circle. 1. 2. y 3. y (0, 1) y (0, 1) (0, 1) (x, y) 135° (−1, 0) (1, 0) (−1, 0) (1, 0) x (0, 0) (0, 0) (−1, 0) 225° (1, 0) (0, 0) x 315° (x, y) (0, −1) 460 Chapter 9 Trigonometric Ratios and Functions (0, −1) (x, y) (0, −1) x 9.1 Right Triangle Trigonometry Essential Question How can you find a trigonometric function of opp. Tangent tan θ = — adj. adj. Cotangent cot θ = — opp. hyp. sec θ = — adj. hyp. csc θ = — opp. Secant te nu adj. cos θ = — hyp. Cosine Cosecant po opp. sin θ = — hyp. hy Sine se Consider one of the acute angles θ of a right triangle. Ratios of a right triangle’s side lengths are used to define the six trigonometric functions, as shown. opposite side an acute angle θ? θ adjacent side Trigonometric Functions of Special Angles Work with a partner. Find the exact values of the sine, cosine, and tangent functions for the angles 30°, 45°, and 60° in the right triangles shown. 60° CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results in constructing arguments. 2 45° 2 1 30° 1 45° 1 3 Exploring Trigonometric Identities Work with a partner. Use the definitions of the trigonometric functions to explain why each trigonometric identity is true. a. sin θ = cos(90° − θ) b. cos θ = sin(90° − θ) 1 c. sin θ = — csc θ 1 d. tan θ = — cot θ Use the definitions of the trigonometric functions to complete each trigonometric identity. e. (sin θ)2 + (cos θ)2 = f. (sec θ)2 − (tan θ)2 = Communicate Your Answer 3. How can you find a trigonometric function of an acute angle θ? 4. Use a calculator to find the lengths x and y of the legs of the right triangle shown. 1 y 25° x Section 9.1 hsnb_alg2_pe_0901.indd 461 Right Triangle Trigonometry 461 2/5/15 1:47 PM Lesson 9.1 What You Will Learn Evaluate trigonometric functions of acute angles. Find unknown side lengths and angle measures of right triangles. Core Vocabul Vocabulary larry Use trigonometric functions to solve real-life problems. sine, p. 462 cosine, p. 462 tangent, p. 462 cosecant, p. 462 secant, p. 462 cotangent, p. 462 The Six Trigonometric Functions Consider a right triangle that has an acute angle θ (the Greek letter theta). The three sides of the triangle are the hypotenuse, the side opposite θ, and the side adjacent to θ. Previous right triangle hypotenuse acute angle Pythagorean Theorem reciprocal complementary angles hypotenuse opposite side Ratios of a right triangle’s side lengths are used to define the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These six functions are abbreviated sin, cos, tan, csc, sec, and cot, respectively. θ adjacent side Core Concept Right Triangle Definitions of Trigonometric Functions Let θ be an acute angle of a right triangle. The six trigonometric functions of θ are defined as shown. REMEMBER The Pythagorean Theorem states that a2 + b2 = c2 for a right triangle with hypotenuse of length c and legs of lengths a and b. a c b opposite sin θ = — hypotenuse adjacent cos θ = — hypotenuse opposite tan θ = — adjacent hypotenuse csc θ = — opposite hypotenuse sec θ = — adjacent adjacent cot θ = — opposite The abbreviations opp., adj., and hyp. are often used to represent the side lengths of the right triangle. Note that the ratios in the second row are reciprocals of the ratios in the first row. 1 csc θ = — sin θ 1 sec θ = — cos θ 1 cot θ = — tan θ Evaluating Trigonometric Functions Evaluate the six trigonometric functions of the angle θ. 5 SOLUTION From the Pythagorean Theorem, the length of the hypotenuse is θ hypotenuse 12 — hyp. = √ 52 + 122 — = √ 169 = 13. Using adj. = 5, opp. = 12, and hyp. = 13, the values of the six trigonometric functions of θ are: 462 Chapter 9 hsnb_alg2_pe_0901.indd 462 opp. 12 sin θ = — = — hyp. 13 5 adj. cos θ = — = — hyp. 13 opp. 12 tan θ = — = — adj. 5 hyp. 13 csc θ = — = — opp. 12 hyp. 13 sec θ = — = — adj. 5 5 adj. cot θ = — = — opp. 12 Trigonometric Ratios and Functions 2/5/15 1:47 PM Evaluating Trigonometric Functions In a right triangle, θ is an acute angle and sin θ = —47 . Evaluate the other five trigonometric functions of θ. SOLUTION Step 1 Draw a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. 7 4 Step 2 Find the length of the adjacent side. By the Pythagorean Theorem, the length of the other leg is adj. = θ 33 — — adj. = √ 72 − 42 = √ 33 . Step 3 Find the values of the remaining five trigonometric functions. 4 hyp. 7 Because sin θ = —, csc θ = — = —. The other values are: 7 opp. 4 — — 4√ 33 4 opp. tan θ = — = — — = — adj. 33 √33 √33 adj. cos θ = — = — hyp. 7 — 7√ 33 7 hyp. sec θ = — = — — = — adj. 33 √33 Monitoring Progress — √33 adj. cot θ = — = — opp. 4 Help in English and Spanish at BigIdeasMath.com Evaluate the six trigonometric functions of the angle θ. 1. 2. 3 θ 3. θ 17 5 2 θ 5 15 4 7 4. In a right triangle, θ is an acute angle and cos θ = — . Evaluate the other five 10 trigonometric functions of θ. The angles 30°, 45°, and 60° occur frequently in trigonometry. You can use the trigonometric values for these angles to find unknown side lengths in special right triangles. Core Concept Trigonometric Values for Special Angles The table gives the values of the six trigonometric functions for the angles 30°, 45°, and 60°. You can obtain these values from the triangles shown. θ 2 30° 45° 1 45° 1 — 45° — 60° √3 — 2 — √2 2 — Section 9.1 hsnb_alg2_pe_0901.indd 463 1 2 30° 3 2 60° 1 sin θ cos θ — √3 2 — tan θ 1 — 2 √3 3 2 1 √2 — √3 sec θ — — — √2 2 — csc θ — 2√3 3 — — — — √3 √2 1 2 √3 — 3 — 2√3 — 3 cot θ — Right Triangle Trigonometry 463 2/5/15 1:47 PM Finding Side Lengths and Angle Measures Finding an Unknown Side Length Find the value of x for the right triangle. 8 SOLUTION 30° x Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x. adj. cos 30° = — hyp. — √3 x —=— 2 8 Write trigonometric equation. Substitute. — 4√ 3 = x Multiply each side by 8. — The length of the side is x = 4√3 ≈ 6.93. Finding all unknown side lengths and angle measures of a triangle is called solving the triangle. Solving right triangles that have acute angles other than 30°, 45°, and 60° may require the use of a calculator. Be sure the calculator is set in degree mode. READING Throughout this chapter, a capital letter is used to denote both an angle of a triangle and its measure. The same letter in lowercase is used to denote the length of the side opposite that angle. Using a Calculator to Solve a Right Triangle Solve △ABC. B c SOLUTION Because the triangle is a right triangle, A and B are complementary angles. So, B = 90° − 28° = 62°. a 28° b = 15 A C Next, write two equations using trigonometric functions, one that involves the ratio of a and 15, and one that involves c and 15. Solve the first equation for a and the second equation for c. opp. tan 28° = — adj. a tan 28° = — 15 15(tan 28°) = a 7.98 ≈ a Write trigonometric equation. Substitute. hyp. sec 28° = — adj. c sec 28° = — 15 1 15 — = c cos 28° ( Solve for the variable. ) 16.99 ≈ c Use a calculator. So, B = 62º, a ≈ 7.98, and c ≈ 16.99. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Find the value of x for the right triangle shown. 6 45° x B c A 464 b Chapter 9 hsnb_alg2_pe_0901.indd 464 a C Solve △ABC using the diagram at the left and the given measurements. 6. B = 45°, c = 5 7. A = 32°, b = 10 8. A = 71°, c = 20 9. B = 60°, a = 7 Trigonometric Ratios and Functions 2/5/15 1:47 PM Solving Real-Life Problems Using Indirect Measurement FINDING AN ENTRY POINT The tangent function is used to find the unknown distance because it involves the ratio of x and 2. You are hiking near a canyon. While standing at A, you measure an angle of 90º between B and C, as shown. You then walk to B and measure an angle of 76° between A and C. The distance between A and B is about 2 miles. How wide is the canyon between A and C? C x SOLUTION x tan 76° = — 2 2(tan 76°) = x Write trigonometric equation. B 76° 2 mi A Multiply each side by 2. 8.0 ≈ x Use a calculator. The width is about 8.0 miles. If you look at a point above you, such as the top of a building, the angle that your line of sight makes with a line parallel to the ground is called the angle of elevation. At the top of the building, the angle between a line parallel to the ground and your line of sight is called the angle of depression. These two angles have the same measure. angle of depression angle of elevation you Using an Angle of Elevation A parasailer is attached to a boat with a rope 72 feet long. The angle of elevation from the boat to the parasailer is 28°. Estimate the parasailer’s height above the boat. SOLUTION Step 1 Draw a diagram that represents the situation. 72 ft 28° h Step 2 Write and solve an equation to find the height h. h sin 28° = — 72 Write trigonometric equation. 72(sin 28°) = h Multiply each side by 72. 33.8 ≈ h Use a calculator. The height of the parasailer above the boat is about 33.8 feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 10. In Example 5, find the distance between B and C. 11. WHAT IF? In Example 6, estimate the height of the parasailer above the boat when the angle of elevation is 38°. Section 9.1 hsnb_alg2_pe_0901.indd 465 Right Triangle Trigonometry 465 2/5/15 1:48 PM Exercises 9.1 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE In a right triangle, the two trigonometric functions of θ that are defined using the lengths of the hypotenuse and the side adjacent to θ are __________ and __________. 2. VOCABULARY Compare an angle of elevation to an angle of depression. 3. WRITING Explain what it means to solve a right triangle. 4. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. What is the cosecant of θ ? 1 What is —? sin θ 6 4 What is the ratio of the side opposite θ to the hypotenuse? θ What is the ratio of the hypotenuse to the side opposite θ? Monitoring Progress and Modeling with Mathematics 12. ANALYZING RELATIONSHIPS Evaluate the six In Exercises 5–10, evaluate the six trigonometric functions of the angle θ. (See Example 1.) 5. 6. θ θ 9 In Exercises 13–18, let θ be an acute angle of a right triangle. Evaluate the other five trigonometric functions of θ. (See Example 2.) 8 12 7. 9 θ 5 7 14. cos θ = — 12 5 7 16. csc θ = — 8 13. sin θ = — 11 6 8. 7 3 15 15. tan θ = —6 14 17. sec θ = — 9 θ 9. trigonometric functions of the 90° − θ angle in Exercises 5–10. Describe the relationships you notice. 16 18. cot θ = — 11 10. 10 θ 14 19. ERROR ANALYSIS Describe and correct the error in finding sin θ of the triangle below. θ 18 26 17 11. REASONING Let θ be an acute angle of a right triangle. Use the two trigonometric functions — 4 √97 tan θ = — and sec θ = — to sketch and label 9 9 the right triangle. Then evaluate the other four trigonometric functions of θ. 466 Chapter 9 hsnb_alg2_pe_0901.indd 466 8 θ 15 ✗ opp. 15 sin θ = — = — hyp. 17 Trigonometric Ratios and Functions 2/5/15 1:48 PM 20. ERROR ANALYSIS Describe and correct the error in finding csc θ, given that θ is an acute angle of a right 7 triangle and cos θ = — . 11 ✗ 1 11 csc θ = — = — 7 cos θ 41. MODELING WITH MATHEMATICS To measure the width of a river, you plant a stake on one side of the river, directly across from a boulder. You then walk 100 meters to the right of the stake and measure a 79° angle between the stake and the boulder. What is the width w of the river? (See Example 5.) Not drawn to scale In Exercises 21–26, find the value of x for the right triangle. (See Example 3.) 21. 23. 79° 9 6 60° x 60° x 100 m 24. 30° 42. MODELING WITH MATHEMATICS Katoomba Scenic Railway in Australia is the steepest railway in the world. The railway makes an angle of about 52° with the ground. The railway extends horizontally about 458 feet. What is the height of the railway? 30° 12 13 x 25. w 22. 43. MODELING WITH MATHEMATICS A person whose x 26. 8 45° 7 x 45° x eye level is 1.5 meters above the ground is standing 75 meters from the base of the Jin Mao Building in Shanghai, China. The person estimates the angle of elevation to the top of the building is about 80°. What is the approximate height of the building? (See Example 6.) 44. MODELING WITH MATHEMATICS The Duquesne USING TOOLS In Exercises 27–32, evaluate the trigonometric function using a calculator. Round your answer to four decimal places. 27. cos 14° 28. tan 31° 29. csc 59° 30. sin 23° 31. cot 6° 32. sec 11° Incline in Pittsburgh, Pennsylvania, has an angle of elevation of 30°. The track has a length of about 800 feet. Find the height of the incline. 45. MODELING WITH MATHEMATICS You are standing on the Grand View Terrace viewing platform at Mount Rushmore, 1000 feet from the base of the monument. Not drawn to scale In Exercises 33–40, solve △ABC using the diagram and the given measurements. (See Example 4.) 1000 ft A b C c a B 33. B = 36°, a = 23 34. A = 27°, b = 9 35. A = 55°, a = 17 36. B = 16°, b = 14 37. A = 43°, b = 31 38. B = 31°, a = 23 39. B = 72°, c = 12.8 40. A = 64°, a = 7.4 a. You look up at the top of Mount Rushmore at an angle of 24°. How high is the top of the monument from where you are standing? Assume your eye level is 5.5 feet above the platform. b. The elevation of the Grand View Terrace is 5280 feet. Use your answer in part (a) to find the elevation of the top of Mount Rushmore. 46. WRITING Write a real-life problem that can be solved using a right triangle. Then solve your problem. Section 9.1 hsnb_alg2_pe_0901.indd 467 b 24° Right Triangle Trigonometry 467 2/5/15 1:48 PM 47. MATHEMATICAL CONNECTIONS The Tropic of Cancer is the circle of latitude farthest north Tropic of of the equator where Cancer the Sun can appear equator directly overhead. It lies 23.5° north of the equator, as shown. 50. PROBLEM SOLVING You measure the angle of elevation from the ground to the top of a building as 32°. When you move 50 meters closer to the building, the angle of elevation is 53°. What is the height of the building? North Pole 23.5° South Pole 51. MAKING AN ARGUMENT Your friend claims it is possible to draw a right triangle so the values of the cosine function of the acute angles are equal. Is your friend correct? Explain your reasoning. a. Find the circumference of the Tropic of Cancer using 3960 miles as the approximate radius of Earth. b. What is the distance between two points on the Tropic of Cancer that lie directly across from each other? 52. THOUGHT PROVOKING Consider a semicircle with a radius of 1 unit, as shown below. Write the values of the six trigonometric functions of the angle θ. Explain your reasoning. 48. HOW DO YOU SEE IT? Use the figure to answer each question. θ y 90° − θ h θ 53. CRITICAL THINKING A procedure for approximating x π based on the work of Archimedes is to inscribe a regular hexagon in a circle. a. Which side is adjacent to θ ? b. Which side is opposite of θ ? c. Does cos θ = sin(90° − θ)? Explain. 30° 1 x 49. PROBLEM SOLVING A passenger in an airplane sees 30° 1 two towns directly to the left of the plane. a. Use the diagram to solve for x. What is the perimeter of the hexagon? 15° 25° 25,000 ft d x b. Show that a regular n-sided polygon inscribed in a circle of radius 1 has a perimeter of 180 ° 2n sin — . n y ⋅ a. What is the distance d from the airplane to the first town? ( ) c. Use the result from part (b) to find an expression in terms of n that approximates π. Then evaluate the expression when n = 50. b. What is the horizontal distance x from the airplane to the first town? c. What is the distance y between the two towns? Explain the process you used to find your answer. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Perform the indicated conversion. (Skills Review Handbook) 54. 5 years to seconds 55. 12 pints to gallons 56. 5.6 meters to millimeters Find the circumference and area of the circle with the given radius or diameter. (Skills Review Handbook) 57. r = 6 centimeters 468 Chapter 9 hsnb_alg2_pe_0901.indd 468 58. r = 11 inches 59. d = 14 feet Trigonometric Ratios and Functions 2/5/15 1:48 PM 9.2 Angles and Radian Measure Essential Question How can you find the measure of an angle in radians? Let the vertex of an angle be at the origin, with one side of the angle on the positive x-axis. The radian measure of the angle is a measure of the intercepted arc length on a circle of radius 1. To convert between degree and radian measure, use the fact that π radians 180° — = 1. Writing Radian Measures of Angles Work with a partner. Write the radian measure of each angle with the given degree measure. Explain your reasoning. a. b. y y 90° radian measure degree measure 60° 120° 135° 45° π 30° 150° 0° 360° x 180° x 210° 225° 315° 330° 240° 270° 300° Writing Degree Measures of Angles Work with a partner. Write the degree measure of each angle with the given radian measure. Explain your reasoning. y degree measure radian measure 7π 9 5π 9 4π 9 2π 9 x 11π 9 13π 14π 9 9 REASONING ABSTRACTLY To be proficient in math, you need to make sense of quantities and their relationships in problem situations. 16π 9 Communicate Your Answer 3. How can you find the measure of an angle in radians? y 4. The figure shows an angle whose measure is 30 radians. What is the measure of the angle in degrees? How many times greater is 30 radians than 30 degrees? Justify your answers. x 30 radians Section 9.2 hsnb_alg2_pe_0902.indd 469 Angles and Radian Measure 469 2/5/15 1:48 PM 9.2 Lesson What You Will Learn Draw angles in standard position. Find coterminal angles. Core Vocabul Vocabulary larry initial side, p. 470 terminal side, p. 470 standard position, p. 470 coterminal, p. 471 radian, p. 471 sector, p. 472 central angle, p. 472 Previous radius of a circle circumference of a circle Use radian measure. Drawing Angles in Standard Position In this lesson, you will expand your study of angles to include angles with measures that can be any real numbers. Core Concept Angles in Standard Position 90° y terminal side In a coordinate plane, an angle can be formed by fixing one ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex. 0° An angle is in standard position when its vertex is at the origin and its initial side lies on the positive x-axis. x 180° vertex initial 360° side 270° The measure of an angle is positive when the rotation of its terminal side is counterclockwise and negative when the rotation is clockwise. The terminal side of an angle can rotate more than 360°. Drawing Angles in Standard Position Draw an angle with the given measure in standard position. a. 240° b. 500° c. −50° b. Because 500° is 140° more than 360°, the terminal side makes one complete rotation 360° counterclockwise plus 140° more. c. Because −50° is negative, the terminal side is 50° clockwise from the positive x-axis. SOLUTION a. Because 240° is 60° more than 180°, the terminal side is 60° counterclockwise past the negative x-axis. y y 240° y 140° x 500° 60° Monitoring Progress x x −50° Help in English and Spanish at BigIdeasMath.com Draw an angle with the given measure in standard position. 1. 65° 470 Chapter 9 hsnb_alg2_pe_0902.indd 470 2. 300° 3. −120° 4. −450° Trigonometric Ratios and Functions 2/5/15 1:48 PM Finding Coterminal Angles STUDY TIP If two angles differ by a multiple of 360°, then the angles are coterminal. In Example 1(b), the angles 500° and 140° are coterminal because their terminal sides coincide. An angle coterminal with a given angle can be found by adding or subtracting multiples of 360°. Finding Coterminal Angles Find one positive angle and one negative angle that are coterminal with (a) −45° and (b) 395°. SOLUTION There are many such angles, depending on what multiple of 360° is added or subtracted. a. −45° + 360° = 315° −45° − 360° = −405° b. 395° − 360° = 35° 395° − 2(360°) = −325° y y −325° 35° 315° −45° x 395° −405° Monitoring Progress x Help in English and Spanish at BigIdeasMath.com Find one positive angle and one negative angle that are coterminal with the given angle. 5. 80° STUDY TIP Notice that 1 radian is approximately equal to 57.3°. 180° = π radians 180° π — = 1 radian 57.3° ≈ 1 radian 6. 230° 8. −135° 7. 740° Using Radian Measure Angles can also be measured in radians. To define a radian, consider a circle with radius r centered at the origin, as shown. One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. y r 1 radian Because the circumference of a circle is 2πr, there are 2π radians in a full circle. So, degree measure and radian measure are related by the equation 360° = 2π radians, or 180° = π radians. r x Core Concept Converting Between Degrees and Radians Degrees to radians Radians to degrees Multiply degree measure by Multiply radian measure by π radians 180° 180° π radians —. —. Section 9.2 hsnb_alg2_pe_0902.indd 471 Angles and Radian Measure 471 2/5/15 1:48 PM Convert Between Degrees and Radians Convert the degree measure to radians or the radian measure to degrees. π b. −— 12 a. 120° READING The unit “radians” is often omitted. For instance, the π measure −— radians may 12 π be written simply as −—. 12 SOLUTION π radians a. 120° = 120 degrees — 180 degrees ( π π 180° b. −— = −— radians — 12 12 π radians ) )( ( 2π =— 3 ) = −15° Concept Summary Degree and Radian Measures of Special Angles The diagram shows equivalent degree and radian measures for special angles from 0° to 360° (0 radians to 2π radians). You may find it helpful to memorize the equivalent degree and radian measures of special angles in the first quadrant and for π 90° = — radians. All other special angles 2 shown are multiples of these angles. Monitoring Progress 5π 6 π 7π 6 y π 2 radian π measure 3 π 4 90° π 120° 60° 6 135° 45° 30° 150° degree 2π 3π 3 4 180° measure 0° 360° 0 x 2π 210° 330° 225° 315° 11π 240° 300° 6 270° 5π 7π 4 4π 4 5π 3π 3 3 2 Help in English and Spanish at BigIdeasMath.com Convert the degree measure to radians or the radian measure to degrees. 5π 9. 135° 10. −40° 11. — 12. −6.28 4 A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle θ of a sector is the angle formed by the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians. Core Concept Arc Length and Area of a Sector The arc length s and area A of a sector with radius r and central angle θ (measured in radians) are as follows. sector r Arc length: s = rθ Area: A = —12 r 2θ 472 Chapter 9 hsnb_alg2_pe_0902.indd 472 central angle θ arc length s Trigonometric Ratios and Functions 2/5/15 1:49 PM Modeling with Mathematics A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field. SOLUTION 1. Understand the Problem You are given the dimensions of a softball field. You are asked to find the length of the outfield fence and the area of the field. outfield fence 200 ft 2. Make a Plan Find the measure of the central angle in radians. Then use the arc length and area of a sector formulas. 90° 3. Solve the Problem 200 ft Step 1 Convert the measure of the central angle to radians. π radians 90° = 90 degrees — 180 degrees ( COMMON ERROR You must write the measure of an angle in radians when using these formulas for the arc length and area of a sector. ) π = — radians 2 Step 2 Find the arc length and the area of the sector. 1 Area: A = —r 2θ 2 Arc length: s = r θ π = 200 — 2 π 1 = — (200)2 — 2 2 = 100π = 10,000π ≈ 314 ≈ 31,416 ( ) ANOTHER WAY Because the central angle is 90°, the sector represents —14 of a circle with a radius of 200 feet. So, s = —14 The length of the outfield fence is about 314 feet. The area of the field is about 31,416 square feet. 4. Look Back To check the area of the field, consider the square formed using the two 200-foot sides. ⋅ 2πr = — ⋅ 2π (200) 1 4 = 100π By drawing the diagonal, you can see that the area of the field is less than the area of the square but greater than one-half of the area of the square. and A = —41 ⋅ πr 2 = —14 ⋅ π (200) = 10,000π. ( ) 2 1 —2 ⋅ (area of square) 1 2 200 ft area of square ? 90° 200 ft ? — (200)2 < 31,416 < 2002 20,000 < 31,416 < 40,000 Monitoring Progress ✓ Help in English and Spanish at BigIdeasMath.com 13. WHAT IF? In Example 4, the outfield fence is 220 feet from home plate. Estimate the length of the outfield fence and the area of the field. Section 9.2 hsnb_alg2_pe_0902.indd 473 Angles and Radian Measure 473 2/5/15 1:49 PM 9.2 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE An angle is in standard position when its vertex is at the __________ and its __________ lies on the positive x-axis. 2. WRITING Explain how the sign of an angle measure determines its direction of rotation. 3. VOCABULARY In your own words, define a radian. 4. WHICH ONE DOESN’T BELONG? Which angle does not belong with the other three? Explain your reasoning. −90° 450° −270° 90° Monitoring Progress and Modeling with Mathematics In Exercises 5–8, draw an angle with the given measure in standard position. (See Example 1.) 5. 110° 6. 450° 7. −900° 8. −10° 22. OPEN-ENDED Using radian measure, give one positive angle and one negative angle that are coterminal with the angle shown. Justify your answers. y In Exercises 9–12, find one positive angle and one negative angle that are coterminal with the given angle. (See Example 2.) 9. 70° x 315° 10. 255° 11. −125° 12. −800° ANALYZING RELATIONSHIPS In Exercises 23–26, match the angle measure with the angle. In Exercises 13–20, convert the degree measure to radians or the radian measure to degrees. (See Example 3.) 23. 600° 13. 40° 14. 315° 25. — 15. −260° 16. −500° π 9 9π 4 24. −— 5π 6 A. 26. −240° B. y y 3π 4 17. — 18. — 19. −5 20. 12 x x 21. WRITING The terminal side of an angle in standard position rotates one-sixth of a revolution counterclockwise from the positive x-axis. Describe how to find the measure of the angle in both degree and radian measures. 474 Chapter 9 hsnb_alg2_pe_0902.indd 474 C. D. y x y x Trigonometric Ratios and Functions 2/5/15 1:49 PM 27. MODELING WITH MATHEMATICS The observation deck of a building forms a sector with the dimensions shown. Find the length of the safety rail and the area of the deck. (See Example 4.) 10 yd 10 yd safety rail 31. PROBLEM SOLVING When a CD player reads information from the outer edge of a CD, the CD spins about 200 revolutions per minute. At that speed, through what angle does a point on the CD spin in one minute? Give your answer in both degree and radian measures. 32. PROBLEM SOLVING You work every Saturday from 90° 28. MODELING WITH MATHEMATICS In the men’s shot put event at the 2012 Summer Olympic Games, the length of the winning shot was 21.89 meters. A shot put must land within a sector having a central angle of 34.92° to be considered fair. 9:00 a.m. to 5:00 p.m. Draw a diagram that shows the rotation completed by the hour hand of a clock during this time. Find the measure of the angle generated by the hour hand in both degrees and radians. Compare this angle with the angle generated by the minute hand from 9:00 a.m. to 5:00 p.m. USING TOOLS In Exercises 33–38, use a calculator to evaluate the trigonometric function. 4π 3 34. sin — 7π 8 35. csc — 10π 11 36. cot −— 37. cot(−14) 38. cos 6 33. cos — ( 65π ) 39. MODELING WITH MATHEMATICS The rear windshield wiper of a car rotates 120°, as shown. Find the area cleared by the wiper. a. The officials draw an arc across the fair landing area, marking the farthest throw. Find the length of the arc. b. All fair throws in the 2012 Olympics landed within a sector bounded by the arc in part (a). What is the area of this sector? 25 in. 120° 14 in. 29. ERROR ANALYSIS Describe and correct the error in converting the degree measure to radians. ✗ ( 180 degrees 24° = 24 degrees —— π radians = 40. MODELING WITH MATHEMATICS A scientist ) 4320 radians — π performed an experiment to study the effects of gravitational force on humans. In order for humans to experience twice Earth’s gravity, they were placed in a centrifuge 58 feet long and spun at a rate of about 15 revolutions per minute. ≈ 1375.1 radians 30. ERROR ANALYSIS Describe and correct the error in finding the area of a sector with a radius of 6 centimeters and a central angle of 40°. ✗ 1 A = — (6)2(40) = 720 cm2 2 a. Through how many radians did the people rotate each second? b. Find the length of the arc through which the people rotated each second. Section 9.2 hsnb_alg2_pe_0902.indd 475 Angles and Radian Measure 475 2/5/15 1:49 PM 41. REASONING In astronomy, the terminator is the day-night line on a planet that divides the planet into daytime and nighttime regions. The terminator moves across the surface of a planet as the planet rotates. It takes about 4 hours for Earth’s terminator to move across the continental United States. Through what angle has Earth rotated during this time? Give your answer in both degree and radian measures. 44. THOUGHT PROVOKING π is an irrational number, which means that it cannot be written as the ratio of two whole numbers. π can, however, be written exactly as a continued fraction, as follows. 1 3 + ———— 1 7 + ——— 1 15 + ——— 1 1 + —— 1 292 + —— 1 1 + —— 1 1+— 1+... terminator Show how to use this continued fraction to obtain a decimal approximation for π. 45. MAKING AN ARGUMENT Your friend claims that when the arc length of a sector equals the radius, the s2 area can be given by A = —. Is your friend correct? 2 Explain. 42. HOW DO YOU SEE IT? Use the graph to find the measure of θ. Explain your reasoning. y 46. PROBLEM SOLVING A spiral staircase has 15 steps. 4 r=4 Each step is a sector with a radius of 42 inches and a π central angle of —. 8 θ x a. What is the length of the arc formed by the outer edge of a step? b. Through what angle would you rotate by climbing the stairs? c. How many square inches of carpeting would you need to cover the 15 steps? 43. MODELING WITH MATHEMATICS A dartboard is divided into 20 sectors. Each sector is worth a point value from 1 to 20 and has shaded regions that double or triple this value. A sector is shown below. Find the areas of the entire sector, the double region, and the triple region. 3 in. 8 3 3 4 in. triple 47. MULTIPLE REPRESENTATIONS There are 60 minutes in 1 degree of arc, and 60 seconds in 1 minute of arc. The notation 50° 30′ 10″ represents an angle with a measure of 50 degrees, 30 minutes, and 10 seconds. 3 in. 8 2 8 in. a. Write the angle measure 70.55° using the notation above. double b. Write the angle measure 110° 45′ 30″ to the nearest hundredth of a degree. Justify your answer. 1 6 5 in. 8 Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the distance between the two points. (Skills Review Handbook) 48. (1, 4), (3, 6) 49. (−7, −13), (10, 8) 50. (−3, 9), (−3, 16) 51. (2, 12), (8, −5) 52. (−14, −22), (−20, −32) 53. (4, 16), (−1, 34) 476 Chapter 9 hsnb_alg2_pe_0902.indd 476 Trigonometric Ratios and Functions 2/5/15 1:49 PM Trigonometric Functions of Any Angle 9.3 Essential Question How can you use the unit circle to define the trigonometric functions of any angle? Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and — r = √ x2 + y2 ≠ 0. The six trigonometric functions of θ are defined as shown. y sin θ = — r r csc θ = —, y ≠ 0 y x cos θ = — r r sec θ = —, x ≠ 0 x y tan θ = —, x ≠ 0 x x cot θ = —, y ≠ 0 y y (x, y) r θ x Writing Trigonometric Functions Work with a partner. Find the sine, cosine, and tangent of the angle θ in standard position whose terminal side intersects the unit circle at the point (x, y) shown. a. ( −1 , 3 2 2 ( y b. (−12 , 12 ( c. y x y x x (0, −1) d. e. y f. y y (−1, 0) x x x ( 12 , − 2 3 ( ( 12 , −12 ( CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. Communicate Your Answer 2. How can you use the unit circle to define the trigonometric functions of any angle? 3. For which angles are each function undefined? Explain your reasoning. a. tangent b. cotangent Section 9.3 hsnb_alg2_pe_0903.indd 477 c. secant d. cosecant Trigonometric Functions of Any Angle 477 2/5/15 1:49 PM 9.3 Lesson What You Will Learn Evaluate trigonometric functions of any angle. Find and use reference angles to evaluate trigonometric functions. Core Vocabul Vocabulary larry unit circle, p. 479 quadrantal angle, p. 479 reference angle, p. 480 Previous circle radius Pythagorean Theorem Trigonometric Functions of Any Angle You can generalize the right-triangle definitions of trigonometric functions so that they apply to any angle in standard position. Core Concept General Definitions of Trigonometric Functions Let θ be an angle in standard position, and let (x, y) be the point where the terminal side of θ intersects the circle x2 + y2 = r2. The six trigonometric functions of θ are defined as shown. y sin θ = — r x cos θ = — r y tan θ = —, x ≠ 0 x y θ (x, y) r r csc θ = —, y ≠ 0 y r sec θ = —, x ≠ 0 x x cot θ = —, y ≠ 0 y x These functions are sometimes called circular functions. Evaluating Trigonometric Functions Given a Point Let (−4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ. y θ (−4, 3) SOLUTION r Use the Pythagorean Theorem to find the length of r. x — r = √ x2 + y2 — = √ (−4)2 + 32 — = √ 25 =5 Using x = −4, y = 3, and r = 5, the values of the six trigonometric functions of θ are: 478 Chapter 9 hsnb_alg2_pe_0903.indd 478 y 3 sin θ = — = — r 5 r 5 csc θ = — = — y 3 4 x cos θ = — = −— r 5 5 r sec θ = — = −— x 4 3 y tan θ = — = −— x 4 4 x cot θ = — = −— y 3 Trigonometric Ratios and Functions 2/5/15 1:49 PM Core Concept The Unit Circle y The circle + = 1, which has center (0, 0) and radius 1, is called the unit circle. The values of sin θ and cos θ are simply the y-coordinate and x-coordinate, respectively, of the point where the terminal side of θ intersects the unit circle. x2 y −r sin θ = — = — = −1. r r The other functions can be evaluated similarly. θ x r=1 y y sin θ = — = — = y r 1 x x cos θ = — = — = x r 1 ANOTHER WAY The general circle x2 + y2 = r2 can also be used to find the six trigonometric functions of θ. The terminal side of θ intersects the circle at (0, −r). So, y2 (x, y) It is convenient to use the unit circle to find trigonometric functions of quadrantal angles. A quadrantal angle is an angle in standard position whose terminal side lies on π an axis. The measure of a quadrantal angle is always a multiple of 90º, or — radians. 2 Using the Unit Circle Use the unit circle to evaluate the six trigonometric functions of θ = 270º. SOLUTION y Step 1 Draw a unit circle with the angle θ = 270º in standard position. θ Step 2 Identify the point where the terminal side of θ intersects the unit circle. The terminal side of θ intersects the unit circle at (0, −1). x Step 3 Find the values of the six trigonometric functions. Let x = 0 and y = −1 to evaluate the trigonometric functions. (0, −1) y −1 sin θ = — = — = −1 r 1 1 r csc θ = — = — = −1 y −1 x 0 cos θ = — = — = 0 r 1 r 1 sec θ = — = — x 0 y −1 tan θ = — = — x 0 0 x cot θ = — = — = 0 y −1 undefined Monitoring Progress undefined Help in English and Spanish at BigIdeasMath.com Evaluate the six trigonometric functions of θ. 1. 2. y (−8, 15) 3. y θ θ x (3, −3) y θ x x (−5, −12) 4. Use the unit circle to evaluate the six trigonometric functions of θ = 180º. Section 9.3 hsnb_alg2_pe_0903.indd 479 Trigonometric Functions of Any Angle 479 2/5/15 1:49 PM Reference Angles Core Concept READING Reference Angle Relationships The symbol θ′ is read as “theta prime.” Let θ be an angle in standard position. The reference angle for θ is the acute angle θ′ formed by the terminal side of θ and the x-axis. The relationship between θ and θ′ is shown below for nonquadrantal angles θ such that 90° < θ < 360° or, π in radians, — < θ < 2π. 2 y θ′ Quadrant IV Quadrant III Quadrant II y y θ θ θ x Degrees: θ ′ = 180° − θ Radians: θ ′ = π − θ θ′ x Degrees: θ ′ = θ − 180° Radians: θ ′ = θ − π θ′ x Degrees: θ ′ = 360° − θ Radians: θ ′ = 2π − θ Finding Reference Angles 5π Find the reference angle θ ′ for (a) θ = — and (b) θ = −130º. 3 SOLUTION a. The terminal side of θ lies in Quadrant IV. So, y x θ′ θ y 5π π θ′ = 2π − — = —. The figure at the right shows 3 3 π 5π θ = — and θ′ = —. 3 3 b. Note that θ is coterminal with 230º, whose terminal side lies in Quadrant III. So, θ′ = 230º − 180º = 50º. The figure at the left shows θ = −130º and θ′ = 50º. θ x θ′ Reference angles allow you to evaluate a trigonometric function for any angle θ. The sign of the trigonometric function value depends on the quadrant in which θ lies. Core Concept Evaluating Trigonometric Functions Use these steps to evaluate a trigonometric function for any angle θ: Step 1 Find the reference angle θ′. Step 2 Evaluate the trigonometric function for θ′. Step 3 Determine the sign of the trigonometric function value from the quadrant in which θ lies. 480 Chapter 9 hsnb_alg2_pe_0903.indd 480 Signs of Function Values Quadrant II sin θ, csc θ : + cos θ , sec θ : − tan θ , cot θ : − Quadrant III sin θ, csc θ : − cos θ , sec θ : − tan θ , cot θ : + Quadrant I sin θ, csc θ : + cos θ , sec θ : + tan θ , cot θ : + y Quadrant IV x sin θ, csc θ : − cos θ , sec θ : + tan θ , cot θ : − Trigonometric Ratios and Functions 2/5/15 1:49 PM Using Reference Angles to Evaluate Functions 17π Evaluate (a) tan(−240º) and (b) csc —. 6 SOLUTION y a. The angle −240º is coterminal with 120º. The reference angle is θ′ = 180º − 120º = 60º. The tangent function θ′ = 60° is negative in Quadrant II, so — x tan(−240º) = −tan 60º = −√ 3 . θ = −240° 5π 17π b. The angle — is coterminal with —. The 6 6 reference angle is y 5π π θ′ = π − — = —. 6 6 The cosecant function is positive in Quadrant II, so INTERPRETING MODELS This model neglects air resistance and assumes that the projectile’s starting and ending heights are the same. θ′= π6 17π θ= 6 17π π csc — = csc — = 2. 6 6 x Solving a Real-Life Problem The horizontal distance d (in feet) traveled by a projectile launched at an angle θ and with an initial speed v (in feet per second) is given by v2 d = — sin 2θ. Model for horizontal distance 32 Estimate the horizontal distance traveled by a golf ball that is hit at an angle of 50° with an initial speed of 105 feet per second. 50° SOLUTION Note that the golf ball is launched at an angle of θ = 50º with initial speed of v = 105 feet per second. v2 d = — sin 2θ 32 1052 = — sin(2 50°) 32 Write model for horizontal distance. ⋅ Substitute 105 for v and 50º for θ. ≈ 339 Use a calculator. The golf ball travels a horizontal distance of about 339 feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Sketch the angle. Then find its reference angle. −7π 9 Evaluate the function without using a calculator. 6. −260° 5. 210° 7. — 15π 4 8. — 11π 4 11. Use the model given in Example 5 to estimate the horizontal distance traveled by a track and field long jumper who jumps at an angle of 20° and with an initial speed of 27 feet per second. 9. cos(−210º) Section 9.3 hsnb_alg2_pe_0903.indd 481 10. sec — Trigonometric Functions of Any Angle 481 2/5/15 1:49 PM Exercises 9.3 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE A(n) ___________ is an angle in standard position whose terminal side lies on an axis. 2. WRITING Given an angle θ in standard position with its terminal side in Quadrant III, explain how you can use a reference angle to find cos θ. Monitoring Progress and Modeling with Mathematics In Exercises 3 –8, evaluate the six trigonometric functions of θ. (See Example 1.) 3. 4. y In Exercises 15–22, sketch the angle. Then find its reference angle. (See Example 3.) y θ θ x x (5, −12) (4, −3) 15. −100° 16. 150° 17. 320° 18. −370° 15π 4 19. — 5π 6 21. −— 5. 6. y y θ (3, 1) terminal side of an angle θ in standard position. Describe and correct the error in finding tan θ. (−6, −8) 7. 8. y θ ✗ y θ x 13π 6 22. −— 23. ERROR ANALYSIS Let (−3, 2) be a point on the θ x x 8π 3 20. — x 3 tan θ = — = −— y 2 24. ERROR ANALYSIS Describe and correct the error in finding a reference angle θ′ for θ = 650°. x (1, −2) ✗ (−12, −9) θ is coterminal with 290°, whose terminal side lies in Quadrant IV. So, θ′ = 290° − 270° = 20°. In Exercises 9–14, use the unit circle to evaluate the six trigonometric functions of θ. (See Example 2.) 9. θ = 0° π 2 10. θ = 540° 7π 2 In Exercises 25–32, evaluate the function without using a calculator. (See Example 4.) 11. θ = — 12. θ = — 25. sec 135° 26. tan 240° 13. θ = −270° 14. θ = −2π 27. sin(−150°) 28. csc(−420°) ( 34π ) 29. tan −— 7π 4 31. cos — 482 Chapter 9 hsnb_alg2_pe_0903.indd 482 ( −83 π ) 30. cot — 11π 6 32. sec — Trigonometric Ratios and Functions 2/5/15 1:49 PM 38. REASONING A Ferris wheel has a radius of 75 feet. In Exercises 33–36, use the model for horizontal distance given in Example 5. You board a car at the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate 255° counterclockwise before the ride temporarily stops. How high above the ground are you when the ride stops? If the radius of the Ferris wheel is doubled, is your height above the ground doubled? Explain your reasoning. 33. You kick a football at an angle of 60° with an initial speed of 49 feet per second. Estimate the horizontal distance traveled by the football. (See Example 5.) 34. The “frogbot” is a robot designed for exploring rough terrain on other planets. It can jump at a 45° angle with an initial speed of 14 feet per second. Estimate the horizontal distance the frogbot can jump on Earth. 39. DRAWING CONCLUSIONS A sprinkler at ground level is used to water a garden. The water leaving the sprinkler has an initial speed of 25 feet per second. a. Use the model for horizontal distance given in Example 5 to complete the table. Angle of sprinkler, θ Horizontal distance water travels, d 30° 35. At what speed must the in-line skater launch himself 35° off the ramp in order to land on the other side of the ramp? 40° 45° 50° 55° 60° 18° 5 ft b. Which value of θ appears to maximize the horizontal distance traveled by the water? Use the model for horizontal distance and the unit circle to explain why your answer makes sense. 36. To win a javelin throwing competition, your last throw must travel a horizontal distance of at least 100 feet. You release the javelin at a 40° angle with an initial speed of 71 feet per second. Do you win the competition? Justify your answer. 37. MODELING WITH MATHEMATICS A rock climber is using a rock climbing treadmill that is 10 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by 110° so that the rock climber is climbing toward the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill? c. Compare the horizontal distance traveled by the water when θ = (45 − k)° with the distance when θ = (45 + k)°, for 0 < k < 45. 40. MODELING WITH MATHEMATICS Your school’s y marching band is performing at halftime during a football game. In the last formation, the band members form a circle 100 feet wide in the center of the field. You start at a point on the circle 100 feet from the goal line, march 300° around the circle, and then walk toward the goal line to exit the field. How far from the goal line are you at the point where you leave the circle? y 5 ft 110° x 300° ? starting position (50, 0) 100 ft 6 ft (x, y) x ? goal line Section 9.3 hsnb_alg2_pe_0903.indd 483 Trigonometric Functions of Any Angle 483 2/5/15 1:49 PM 41. ANALYZING RELATIONSHIPS Use symmetry and 46. MAKING AN ARGUMENT Your friend claims that the given information to label the coordinates of the other points corresponding to special angles on the unit circle. (0, 1) y ( 12 , 23 ( 90° 60° 45° 30° 120° 135° 150° 0° 360° 330° 315° 300° 270° 180° 210° 225° 240° the only solution to the trigonometric equation — tan θ = √3 is θ = 60°. Is your friend correct? Explain your reasoning. 47. PROBLEM SOLVING When two atoms in a molecule ( 22 , 22 ( ( 23 , 12 ( are bonded to a common atom, chemists are interested in both the bond angle and the lengths of the bonds. An ozone molecule is made up of two oxygen atoms bonded to a third oxygen atom, as shown. x (1, 0) y (x, y) d 128 pm 117° (0, 0) 42. THOUGHT PROVOKING Use the interactive unit circle tool at BigIdeasMath.com to describe all values of θ for each situation. x 128 pm (128, 0) a. In the diagram, coordinates are given in picometers (pm). (Note: 1 pm = 10−12 m) Find the coordinates (x, y) of the center of the oxygen atom in Quadrant II. a. sin θ > 0, cos θ < 0, and tan θ > 0 b. sin θ > 0, cos θ < 0, and tan θ < 0 b. Find the distance d (in picometers) between the centers of the two unbonded oxygen atoms. 43. CRITICAL THINKING Write tan θ as the ratio of two other trigonometric functions. Use this ratio to explain why tan 90° is undefined but cot 90° = 0. 48. MATHEMATICAL CONNECTIONS The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point P in the diagram equals the degree measure of arc PE. At what latitude θ is the circumference of the circle of latitude at P half the distance around the equator? 44. HOW DO YOU SEE IT? Determine whether each of the six trigonometric functions of θ is positive, negative, or zero. Explain your reasoning. y θ circle of latitude x O 45. USING STRUCTURE A line with slope m passes through the origin. An angle θ in standard position has a terminal side that coincides with the line. Use a trigonometric function to relate the slope of the line to the angle. Maintaining Mathematical Proficiency P C θ D E equator Reviewing what you learned in previous grades and lessons Find all real zeros of the polynomial function. (Section 4.6) 49. f (x) = x4 + 2x3 + x2 + 8x − 12 50. f (x) = x5 + 4x4 − 14x3 − 14x2 − 15x − 18 Graph the function. (Section 4.8) 51. f (x) = 2(x + 3)2(x − 1) 484 Chapter 9 hsnb_alg2_pe_0903.indd 484 1 52. f (x) = —3 (x − 4)(x + 5)(x + 9) 53. f (x) = x2(x + 1)3(x − 2) Trigonometric Ratios and Functions 2/5/15 1:49 PM 9.4 Graphing Sine and Cosine Functions Essential Question What are the characteristics of the graphs of the sine and cosine functions? Graphing the Sine Function Work with a partner. a. Complete the table for y = sin x, where x is an angle measure in radians. x −2π 7π −— 4 π 4 — 3π − 5π −π − 3π −π −— — — — 2 4 4 2 π −— 4 0 2π — y = sin x — x π 2 3π 4 — π 5π 4 — 3π 2 — 7π 4 — 9π 4 y = sin x b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to sketch the graph of y = sin x. y 1 −2π π −3 2 −π π − 2 π 2 π 3π 2 2π 5π x 2 −1 c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over −2π ≤ x ≤ 2π. Is the sine function even, odd, or neither? Graphing the Cosine Function Work with a partner. a. Complete a table for y = cos x using the same values of x as those used in Exploration 1. b. Plot the points (x, y) from part (a) and sketch the graph of y = cos x. LOOKING FOR STRUCTURE To be proficient in math, you need to look closely to discern a pattern or structure. c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over −2π ≤ x ≤ 2π. Is the cosine function even, odd, or neither? Communicate Your Answer 3. What are the characteristics of the graphs of the sine and cosine functions? 4. Describe the end behavior of the graph of y = sin x. Section 9.4 hsnb_alg2_pe_0904.indd 485 Graphing Sine and Cosine Functions 485 2/5/15 1:50 PM 9.4 Lesson What You Will Learn Explore characteristics of sine and cosine functions. Stretch and shrink graphs of sine and cosine functions. Core Vocabul Vocabulary larry amplitude, p. 486 periodic function, p. 486 cycle, p. 486 period, p. 486 phase shift, p. 488 midline, p. 488 Previous transformations x-intercept Translate graphs of sine and cosine functions. Reflect graphs of sine and cosine functions. Exploring Characteristics of Sine and Cosine Functions In this lesson, you will learn to graph sine and cosine functions. The graphs of sine and cosine functions are related to the graphs of the parent functions y = sin x and y = cos x, which are shown below. x 3π −2π − — 2 −π π −— 2 0 — π 2 π — 3π 2 2π y = sin x 0 1 0 −1 0 1 0 −1 0 y = cos x 1 0 −1 0 1 0 −1 0 1 y maximum value: 1 y = sin x 1 amplitude: 1 range: −1 ≤ y ≤ 1 − 3π −π 2 π − 2 π 2 −1 range: −1 ≤ y ≤ 1 3π 2 2π x period: 2π minimum value: −1 maximum m value: 1 y = cos x π y amplitude: 1 − 2π − 3π −π 2 π − 2 minimum value: −1 −1 π 2 π 3π 2 2π x period: 2π Core Concept Characteristics of y = sin x and y = cos x • The domain of each function is all real numbers. • The range of each function is −1 ≤ y ≤ 1. So, the minimum value of each function is −1 and the maximum value is 1. • The amplitude of the graph of each function is one-half of the difference of the maximum value and the minimum value, or —12 [1 − (−1)] = 1. • Each function is periodic, which means that its graph has a repeating pattern. The shortest repeating portion of the graph is called a cycle. The horizontal length of each cycle is called the period. Each graph shown above has a period of 2π. • The x-intercepts for y = sin x occur when x = 0, ±π, ±2π, ±3π, . . .. π 3π 5π 7π • The x-intercepts for y = cos x occur when x = ± —, ± —, ± —, ± —, . . .. 2 2 2 2 486 Chapter 9 hsnb_alg2_pe_0904.indd 486 Trigonometric Ratios and Functions 2/5/15 1:50 PM Stretching and Shrinking Sine and Cosine Functions REMEMBER ⋅ The graph of y = a f (x) is a vertical stretch or shrink of the graph of y = f (x) by a factor of a. The graphs of y = a sin bx and y = a cos bx represent transformations of their parent functions. The value of a indicates a vertical stretch (a > 1) or a vertical shrink (0 < a < 1) and changes the amplitude of the graph. The value of b indicates a horizontal stretch (0 < b < 1) or a horizontal shrink (b > 1) and changes the period of the graph. y = a sin bx y = a cos bx 1 horizontal stretch or shrink by a factor of — b vertical stretch or shrink by a factor of a The graph of y = f (bx) is a horizontal stretch or shrink of the graph of 1 y = f (x) by a factor of —. b Core Concept Amplitude and Period The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where a and b are nonzero real numbers, are as follows: 2π Period = — ∣b∣ Amplitude = ∣ a ∣ 2π Each graph below shows five key points that partition the interval 0 ≤ x ≤ — into b four equal parts. You can use these points to sketch the graphs of y = a sin bx and y = a cos bx. The x-intercepts, maximum, and minimum occur at these points. y ( 14 ∙ 2bπ , a( y y = a sin bx (0, a) ( 2bπ , 0( (0, 0) y = a cos bx x ( 12 ∙ 2bπ , 0( ( 2bπ , a( ( 14 ∙ 2bπ , 0( ( 34 ∙ 2bπ , 0( x ( 12 ∙ 2bπ , −a( ( 34 ∙ 2bπ , −a( Graphing a Sine Function Identify the amplitude and period of g(x) = 4 sin x. Then graph the function and describe the graph of g as a transformation of the graph of f (x) = sin x. REMEMBER A vertical stretch of a graph does not change its x-intercept(s). So, it makes sense that the x-intercepts of g(x) = 4 sin x and f (x) = sin x are the same. 4 g − f π 4 9π 4 SOLUTION The function is of the form g(x) = a sin bx where a = 4 and b = 1. So, the amplitude 2π 2π is a = 4 and the period is — = — = 2π. b 1 ( ) π 1 Maximum: — 2π, 4 = —, 4 4 2 (⋅ ) ( ) 3 3π Minimum: ( ⋅ 2π, −4 ) = ( , −4 ) 4 2 — −4 ⋅ 1 Intercepts: (0, 0); — 2π, 0 = (π, 0); (2π, 0) 2 y 4 π 2 3π 2 x — The graph of g is a vertical stretch by a factor of 4 of the graph of f. Section 9.4 hsnb_alg2_pe_0904.indd 487 Graphing Sine and Cosine Functions 487 2/5/15 1:50 PM Graphing a Cosine Function 1 Identify the amplitude and period of g(x) = — cos 2πx. Then graph the function and 2 describe the graph of g as a transformation of the graph of f (x) = cos x. SOLUTION 1 The function is of the form g(x) = a cos bx where a = — and b = 2π. So, the 2 2π 2π 1 amplitude is a = — and the period is — = — = 1. 2 b 2π STUDY TIP After you have drawn one complete cycle of the graph in Example 2 on the interval 0 ≤ x ≤ 1, you can extend the graph by repeating the cycle as many times as desired to the left and right of 0 ≤ x ≤ 1. ( ) ( )( ⋅ ) ( ) 1 1 Maximums: ( 0, ); ( 1, ) 2 2 1 1 1 1 Minimum: ( ⋅ 1, − ) = ( , − ) 2 2 2 2 ⋅ 1 1 3 3 Intercepts: — 1, 0 = — , 0 ; — 1, 0 = — , 0 4 4 4 4 — — y — — 1 — — 1 2 x −1 1 The graph of g is a vertical shrink by a factor of — and a horizontal shrink by a 2 1 factor of — of the graph of f. 2π Monitoring Progress Help in English and Spanish at BigIdeasMath.com Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function. REMEMBER 1 The graph of y = f (x) + k is a vertical translation of the graph of y = f (x). The graph of y = f (x − h) is a horizontal translation of the graph of y = f (x). 1. g(x) = —4 sin x 2. g(x) = cos 2x 3. g(x) = 2 sin πx 1 1 4. g(x) = —3 cos —2 x Translating Sine and Cosine Functions The graphs of y = a sin b(x − h) + k and y = a cos b(x − h) + k represent translations of y = a sin bx and y = a cos bx. The value of k indicates a translation up (k > 0) or down (k < 0). The value of h indicates a translation left (h < 0) or right (h > 0). A horizontal translation of a periodic function is called a phase shift. Core Concept Graphing y = a sin b(x − h) + k and y = a cos b(x − h) + k To graph y = a sin b(x − h) + k or y = a cos b(x − h) + k where a > 0 and b > 0, follow these steps: 2π Step 1 Identify the amplitude a, the period —, the horizontal shift h, and the b vertical shift k of the graph. Step 2 Draw the horizontal line y = k, called the midline of the graph. Step 3 Find the five key points by translating the key points of y = a sin bx or y = a cos bx horizontally h units and vertically k units. Step 4 Draw the graph through the five translated key points. 488 Chapter 9 hsnb_alg2_pe_0904.indd 488 Trigonometric Ratios and Functions 2/5/15 1:50 PM Graphing a Vertical Translation Graph g(x) = 2 sin 4x + 3. LOOKING FOR STRUCTURE The graph of g is a translation 3 units up of the graph of f (x) = 2 sin 4x. So, add 3 to the y-coordinates of the five key points of f. SOLUTION Step 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude: a = 2 Horizontal shift: h = 0 2π 2π π Period: — = — = — b 4 2 Vertical shift: k = 3 Step 2 Draw the midline of the graph, y = 3. Step 3 Find the five key points. π π π π On y = k: (0, 0 + 3) = (0, 3); —, 0 + 3 = —, 3 ; —, 0 + 3 = —, 3 4 4 2 2 ( π π Maximum: ( , 2 + 3 ) = ( , 5 ) 8 8 — ) ( )( — y 5 3π 3π Minimum: —, −2 + 3 = —, 1 8 8 ( ) ( ) ) ( ) 1 Step 4 Draw the graph through the key points. π 4 −1 π 2 x Graphing a Horizontal Translation 1 Graph g(x) = 5 cos — (x − 3π). 2 LOOKING FOR STRUCTURE The graph of g is a translation 3π units right of the graph of f (x) = 5 cos —12 x. So, add 3π to the x-coordinates of the five key points of f. SOLUTION Step 1 Identify the amplitude, period, horizontal shift, and vertical shift. Horizontal shift: h = 3π Amplitude: a = 5 2π 2π Period: — = — = 4π b 1 — 2 Vertical shift: k = 0 Step 2 Draw the midline of the graph. Because k = 0, the midline is the x-axis. Step 3 Find the five key points. y On y = k: (π + 3π, 0) = (4π, 0); (3π + 3π, 0) = (6π, 0) 6 2 Maximums: (0 + 3π, 5) = (3π, 5); (4π + 3π, 5) = (7π, 5) −2 Minimum: (2π + 3π, −5) = (5π, −5) 3π 5π 7π 9π −6 Step 4 Draw the graph through the key points. Monitoring Progress x π Help in English and Spanish at BigIdeasMath.com Graph the function. 5. g(x) = cos x + 4 Section 9.4 hsnb_alg2_pe_0904.indd 489 1 2 ( π 2 6. g(x) = — sin x − — ) 7. g(x) = sin(x + π) − 1 Graphing Sine and Cosine Functions 489 2/5/15 1:50 PM Reflecting Sine and Cosine Functions You have graphed functions of the form y = a sin b(x − h) + k and y = a cos b(x − h) + k, where a > 0 and b > 0. To see what happens when a < 0, consider the graphs of y = −sin x and y = −cos x. y y = −sin x 1 REMEMBER This result makes sense because the graph of y = −f (x) is a reflection in the x-axis of the graph of y = f (x). π 2 −1 1 (2π, 0) (0, 0) y = −cos x y ( 32π , 1( (π, 1) ( 32π , 0( ( π2, 0( π x (π, 0) (0, −1) ( π2 , −1( x 2π (2π, −1) The graphs are reflections of the graphs of y = sin x and y = cos x in the x-axis. In general, when a < 0, the graphs of y = a sin b(x − h) + k and y = a cos b(x − h) + k are reflections of the graphs of y = ∣ a ∣ sin b(x − h) + k and y = ∣ a ∣ cos b(x − h) + k, respectively, in the midline y = k. Graphing a Reflection π 2 Graph g(x) = −2 sin — x − — . 3 2 ) ( SOLUTION Step 1 Identify the amplitude, period, horizontal shift, and vertical shift. π Horizontal shift: h = — 2 Amplitude: ∣ a ∣ = ∣ −2 ∣ = 2 2π 2π Period: — = — = 3π b 2 — 3 Vertical shift: k = 0 Step 2 Draw the midline of the graph. Because k = 0, the midline is the x-axis. π 2 Step 3 Find the five key points of f (x) = ∣ −2 ∣ sin — x − — . 3 2 π π π 3π π 7π On y = k: 0 + —, 0 = —, 0 ; — + —, 0 = (2π, 0); 3π + —, 0 = —, 0 2 2 2 2 2 2 ( ) ( )( In Example 5, the maximum value and minimum value of f are the minimum value and maximum value, respectively, of g. ) ) 3π π 5π 9π π 11π , −2 ) Maximum: ( + , 2) = ( , 2) Minimum: ( + , −2 ) = ( 4 2 4 4 2 4 ( STUDY TIP ) — — — — Step 4 Reflect the graph. Because a < 0, the graph is reflected in the midline 5π 5π y = 0. So, —, 2 becomes —, −2 4 4 11π 11π and —, −2 becomes —, 2 . 4 4 ( ) ( ) ( ( ) ( ( — — y ) ) 1 −1 π 3π x Step 5 Draw the graph through the key points. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. ( π 2 8. g(x) = −cos x + — 490 Chapter 9 hsnb_alg2_pe_0904.indd 490 ) 1 2 9. g(x) = −3 sin — x + 2 10. g(x) = −2 cos 4x − 1 Trigonometric Ratios and Functions 2/5/15 1:50 PM 9.4 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE The shortest repeating portion of the graph of a periodic function is called a(n) _________. 1 2. WRITING Compare the amplitudes and periods of the functions y = —2 cos x and y = 3 cos 2x. 3. VOCABULARY What is a phase shift? Give an example of a sine function that has a phase shift. 4. VOCABULARY What is the midline of the graph of the function y = 2 sin 3(x + 1) − 2? Monitoring Progress and Modeling with Mathematics USING STRUCTURE In Exercises 5–8, determine whether the graph represents a periodic function. If so, identify the period. y 5. y 6. 1 1 x 2 4 π 2 x In Exercises 13–20, identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function. (See Examples 1 and 2.) 13. g(x) = 3 sin x 14. g(x) = 2 sin x 15. g(x) = cos 3x 16. g(x) = cos 4x 17. g(x) = sin 2π x 18. g(x) = 3 sin 2x 1 19. g(x) = —3 cos 4x 7. 8. y y 21. ANALYZING EQUATIONS Which functions have an 4 1 1 20. g(x) = —2 cos 4πx amplitude of 4 and a period of 2? 2 10 x A y = 4 cos 2x ○ B y = −4 sin πx ○ −1 2 4 6 x C y = 2 sin 4x ○ In Exercises 9–12, identify the amplitude and period of the graph of the function. y 9. 10. D y = 4 cos πx ○ 22. WRITING EQUATIONS Write an equation of the form y = a sin bx, where a > 0 and b > 0, so that the graph has the given amplitude and period. y 0.5 1 2π 11. x 1 12. y 2 x y 2 π 2 π x π 4π 7π x −4 Section 9.4 hsnb_alg2_pe_0904.indd 491 b. amplitude: 10 period: 4 c. amplitude: 2 period: 2π d. amplitude: —12 period: 3π 23. MODELING WITH MATHEMATICS The motion 4 −2 a. amplitude: 1 period: 5 of a pendulum can be modeled by the function d = 4 cos 8π t, where d is the horizontal displacement (in inches) of the pendulum relative to its position at rest and t is the time (in seconds). Find and interpret the period and amplitude in the context of this situation. Then graph the function. Graphing Sine and Cosine Functions 491 2/5/15 1:50 PM 24. MODELING WITH MATHEMATICS A buoy bobs up and down as waves go past. The vertical displacement y (in feet) of the buoy with respect to sea level can be π modeled by y = 1.75 cos —t, where t is the time 3 (in seconds). Find and interpret the period and amplitude in the context of the problem. Then graph the function. USING STRUCTURE In Exercises 37– 40, describe the transformation of the graph of f represented by the function g. ( π2 ) π f (x) = sin x, g(x) = 3 sin( x + ) − 2 4 37. f (x) = cos x, g(x) = 2 cos x − — + 1 38. — 39. f (x) = sin x, g(x) = sin 3(x + 3π) − 5 40. f (x) = cos x, g(x) = cos 6(x − π) + 9 In Exercises 41–48, graph the function. (See Example 5.) 41. g(x) = −cos x + 3 42. g(x) = −sin x − 5 1 2 43. g(x) = −sin —x − 2 In Exercises 25–34, graph the function. (See Examples 3 and 4.) 25. g(x) = sin x + 2 π 2 ( 27. g(x) = cos x − — ) ( π 4 29. g(x) = 2 cos x − 1 45. g(x) = −sin(x − π) + 4 46. g(x) = −cos(x + π) − 2 26. g(x) = cos x − 4 28. g(x) = sin x + — ) 30. g(x) = 3 sin x + 1 31. g(x) = sin 2(x + π) ( π4 ) π g(x) = −5 sin( x − ) + 3 2 47. g(x) = −4 cos x + — − 1 48. — 49. USING EQUATIONS Which of the following is a point where the maximum value of the graph of π y = −4 cos x − — occurs? 2 π π −—, 4 A B —, 4 ○ ○ 2 2 32. g(x) = cos 2(x − π) ( 1 33. g(x) = sin —(x + 2π) + 3 2 ( 1 2 ) ) ( ) D (π, 4) ○ C (0, 4) ○ 34. g(x) = cos —(x − 3π) − 5 35. ERROR ANALYSIS Describe and correct the error in 50. ANALYZING RELATIONSHIPS Match each function with its graph. Explain your reasoning. 2 finding the period of the function y = sin —x. 3 ✗ 44. g(x) = −cos 2x + 1 2 ∣= 1 ∣b∣ ∣ — 3 Period: = — 2π — 2π — 3π a. y = 3 + sin x b. y = −3 + cos x π c. y = sin 2 x − — 2 ( A. π d. y = cos 2 x − — 2 ) ( B. y π 2 36. ERROR ANALYSIS Describe and correct the error in ( ✗ ) −1 hsnb_alg2_pe_0904.indd 492 x y C. Maximum: ( ( —14 ⋅2π ) − —π2, 2 ) = ( —π2 − —π2, 2 ) Chapter 9 π 1 π = (0, 2) 492 y 4 1 determining the point where the maximum value of π the function y = 2 sin x − — occurs. 2 −1 ) D. π 2π x 2π x y 1 x π 2 π −4 Trigonometric Ratios and Functions 2/5/15 1:50 PM WRITING EQUATIONS In Exercises 51–54, write a rule for g that represents the indicated transformations of the graph of f. 57. USING TOOLS The average wind speed s (in miles per hour) in the Boston Harbor can be approximated by π s = 3.38 sin — (t + 3) + 11.6 180 51. f (x) = 3 sin x; translation 2 units up and π units right where t is the time in days and t = 0 represents January 1. Use a graphing calculator to graph the function. On which days of the year is the average wind speed 10 miles per hour? Explain your reasoning. 52. f (x) = cos 2πx; translation 4 units down and 3 units left 1 53. f (x) = —3 cos πx; translation 1 unit down, followed by a reflection in the line y = −1 1 58. USING TOOLS The water depth d (in feet) for the Bay 3 54. f (x) = —2 sin 6x; translation —2 units down and 1 unit π of Fundy can be modeled by d = 35 − 28 cos —t, 6.2 where t is the time in hours and t = 0 represents midnight. Use a graphing calculator to graph the function. At what time(s) is the water depth 7 feet? Explain. 3 right, followed by a reflection in the line y = −—2 55. MODELING WITH MATHEMATICS The height h (in feet) of a swing above the ground can be modeled by the function h = −8 cos θ + 10, where the pivot is 10 feet above the ground, the rope is 8 feet long, and θ is the angle that the rope makes with the vertical. Graph the function. What is the height of the swing when θ is 45°? 8 ft 10 − h 8 ft θ high tide low tide 10 ft h 59. MULTIPLE REPRESENTATIONS Find the average rate of Front view change of each function over the interval 0 < x < π. Side view a. y = 2 cos x 56. DRAWING A CONCLUSION In a particular region, the b. population L (in thousands) of lynx (the predator) and the population H (in thousands) of hares (the prey) can be modeled by the equations x f (x) = −cos x π L = 11.5 + 6.5 sin — t 5 π H = 27.5 + 17.5 cos — t 5 c. 1 b. Use the figure to explain how the changes in the two populations appear to be related. Population (thousands) 20 2π −1 0 1 0 −1 f x b. Graph each function. L 16 c. Describe the transformations of the graphs of the parent functions. t Time (years) Section 9.4 hsnb_alg2_pe_0904.indd 493 3π 2 a. Construct a table of values for each equation using the quadrantal angles in the interval −2π ≤ x ≤ 2π. H 40 12 — y = cos(−x). y 8 π 60. REASONING Consider the functions y = sin(−x) and Animal Populations 4 π 2 π a. Determine the ratio of hares to lynx when t = 0, 2.5, 5, and 7.5 years. 0 — y where t is the time in years. 0 0 Graphing Sine and Cosine Functions 493 2/5/15 1:50 PM 61. MODELING WITH MATHEMATICS You are riding a 66. THOUGHT PROVOKING Use a graphing calculator to find a function of the form y = sin b1x + cos b2x whose graph matches that shown below. Ferris wheel that turns for 180 seconds. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the equation π h = 85 sin —(t − 10) + 90. 20 a. Graph the function. y −6 −4 b. How many cycles does the Ferris wheel make in 180 seconds? −2 2 4 6x −2 c. What are your maximum and minimum heights? 67. PROBLEM SOLVING For a person at rest, the blood pressure P (in millimeters of mercury) at time t (in seconds) is given by the function 62. HOW DO YOU SEE IT? Use the graph to answer each question. 8π P = 100 − 20 cos —t. 3 Graph the function. One cycle is equivalent to one heartbeat. What is the pulse rate (in heartbeats per minute) of the person? y 6 −π π x −6 a. Does the graph represent a function of the form f(x) = a sin bx or f(x) = a cos bx? Explain. b. Identify the maximum value, minimum value, period, and amplitude of the function. 68. PROBLEM SOLVING The motion of a spring can be modeled by y = A cos kt, where y is the vertical displacement (in feet) of the spring relative to its position at rest, A is the initial displacement (in feet), k is a constant that measures the elasticity of the spring, and t is the time (in seconds). 63. FINDING A PATTERN Write an expression in terms of the integer n that represents all the x-intercepts of the graph of the function y = cos 2x. Justify your answer. 64. MAKING AN ARGUMENT Your friend states that for a. You have a spring whose motion can be modeled by the function y = 0.2 cos 6t. Find the initial displacement and the period of the spring. Then graph the function. functions of the form y = a sin bx and y = a cos bx, the values of a and b affect the x-intercepts of the graph of the function. Is your friend correct? Explain. b. When a damping force is applied to the spring, the motion of the spring can be modeled by the function y = 0.2e−4.5t cos 4t. Graph this function. What effect does damping have on the motion? 65. CRITICAL THINKING Describe a transformation of the graph of f (x) = sin x that results in the graph of g(x) = cos x. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Simplify the rational expression, if possible. (Section 7.3) x2 + x − 6 x+3 69. — x3 − 2x2 − 24x x − 2x − 24 70. —— 2 x2 − 4x − 5 x + 4x − 5 71. — 2 x2 − 16 x + x − 20 72. — 2 Find the least common multiple of the expressions. (Section 7.4) 73. 2x, 2(x − 5) 494 Chapter 9 hsnb_alg2_pe_0904.indd 494 74. x2 − 4, x + 2 75. x2 + 8x + 12, x + 6 Trigonometric Ratios and Functions 2/5/15 1:50 PM 9.1–9.4 What Did You Learn? Core Vocabulary sine, p. 462 cosine, p. 462 tangent, p. 462 cosecant, p. 462 secant, p. 462 cotangent, p. 462 initial side, p. 470 terminal side, p. 470 standard position, p. 470 coterminal, p. 471 radian, p. 471 sector, p. 472 central angle, p. 472 unit circle, p. 479 quadrantal angle, p. 479 reference angle, p. 480 amplitude, p. 486 periodic function, p. 486 cycle, p. 486 period, p. 486 phase shift, p. 488 midline, p. 488 Core Concepts Section 9.1 Right Triangle Definitions of Trigonometric Functions, p. 462 Trigonometric Values for Special Angles, p. 463 Section 9.2 Angles in Standard Position, p. 470 Converting Between Degrees and Radians, p. 471 Degree and Radian Measures of Special Angles, p. 472 Arc Length and Area of a Sector, p. 472 Section 9.3 General Definitions of Trigonometric Functions, p. 478 The Unit Circle, p. 479 Reference Angle Relationships, p. 480 Evaluating Trigonometric Functions, p. 480 Section 9.4 Characteristics of y = sin x and y = cos x, p. 486 Amplitude and Period, p. 487 Graphing y = a sin b(x − h) + k and y = a cos b(x − h) + k, p. 488 Mathematical Practices 1. Make a conjecture about the horizontal distances traveled in part (c) of Exercise 39 on page 483. 2. Explain why the quantities in part (a) of Exercise 56 on page 493 make sense in the context of the situation. Study Skills Form a Final Exam Study Group Form a study group several weeks before the final exam. The intent of this group is to review what you have already learned while continuing to learn new material. 495 hsnb_alg2_pe_09mc.indd 495 2/5/15 1:46 PM 9.1–9.4 Quiz 2 1. In a right triangle, θ is an acute angle and sin θ = —7 . Evaluate the other five trigonometric functions of θ. (Section 9.1) Find the value of x for the right triangle. (Section 9.1) 2. 3. 60° 30° x 8 4. 12 27 x 49° x Draw an angle with the given measure in standard position. Then find one positive angle and one negative angle that are coterminal with the given angle. (Section 9.2) 5π 6 5. 40° 7. −960° 6. — Convert the degree measure to radians or the radian measure to degrees. (Section 9.2) 3π 10 9. −60° 8. — 10. 72° Evaluate the six trigonometric functions of θ. (Section 9.3) 11. 12. y 13. y y θ = π2 θ x 2π θ= 3 x x (−2, −6) 14. Identify the amplitude and period of g(x) = 3 sin x. Then graph the function and describe the graph of g as a transformation of the graph of f (x) = sin x. (Section 9.4) 15. Identify the amplitude and period of g(x) = cos 5πx + 3. Then graph the function and describe the graph of g as a transformation of the graph of f(x) = cos x. (Section 9.4) 16. You are flying a kite at an angle of 70°. You have let out a total of 400 feet of string and are holding the reel steady 4 feet above the ground. (Section 9.1) a. How high above the ground is the kite? b. A friend watching the kite estimates that the angle of elevation to the kite is 85°. How far from your friend are you standing? 400 ft Not drawn to scale 70° 85° 4 ft 17. The top of the Space Needle in Seattle, Washington, is a revolving, circular restaurant. The restaurant has a radius of 47.25 feet and makes one complete revolution in about an hour. You have dinner at a window table from 7:00 p.m. to 8:55 p.m. Compare the distance you revolve with the distance of a person seated 5 feet away from the windows. (Section 9.2) 496 Chapter 9 hsnb_alg2_pe_09mc.indd 496 Trigonometric Ratios and Functions 2/5/15 1:46 PM 9.5 Graphing Other Trigonometric Functions Essential Question What are the characteristics of the graph of the tangent function? Graphing the Tangent Function Work with a partner. a. Complete the table for y = tan x, where x is an angle measure in radians. x π −— 2 π −— 3 π −— 4 π −— 6 0 — π 6 — π 4 — π 3 — 2π 3 — 3π 4 — 5π 6 π — 7π 6 — 5π 4 — 4π 3 — 3π 2 — π 2 y = tan x x — 5π 3 y = tan x b. The graph of y = tan x has vertical asymptotes at x-values where tan x is undefined. Plot the points (x, y) from part (a). Then use the asymptotes to sketch the graph of y = tan x. y 6 4 2 − π 2 π 2 π 3π 2 x −2 −4 −6 MAKING SENSE OF PROBLEMS To be proficient in math, you need to consider analogous problems and try special cases of the original problem in order to gain insight into its solution. c. For the graph of y = tan x, identify the asymptotes, the x-intercepts, and the π 3π intervals for which the function is increasing or decreasing over −— ≤ x ≤ —. 2 2 Is the tangent function even, odd, or neither? Communicate Your Answer 2. What are the characteristics of the graph of the tangent function? π 2 Section 9.5 hsnb_alg2_pe_0905.indd 497 3π 2 3. Describe the asymptotes of the graph of y = cot x on the interval −— < x < —. Graphing Other Trigonometric Functions 497 2/5/15 1:51 PM 9.5 Lesson What You Will Learn Explore characteristics of tangent and cotangent functions. Graph tangent and cotangent functions. Core Vocabul Vocabulary larry Graph secant and cosecant functions. Previous asymptote period amplitude x-intercept transformations Exploring Tangent and Cotangent Functions The graphs of tangent and cotangent functions are related to the graphs of the parent functions y = tan x and y = cot x, which are graphed below. x π −— 2 π x approaches −— 2 −1.57 −1.5 π −— 4 0 — −1 0 1 y = tan x Undef. −1256 −14.10 π 4 tan x approaches −∞ sin x Because tan x = —, tan x cos x is undefined for x-values at which cos x = 0, such as π x = ± — ≈ ±1.571. 2 The table indicates that the graph has asymptotes at these values. The table represents one cycle of the graph, so the period of the graph is π. You can use a similar approach to graph y = cot x. Because cos x cot x = —, cot x is undefined for sin x x-values at which sin x = 0, which are multiples of π. The graph has asymptotes at these values. The period of the graph is also π. 3π 2 1.5 1.57 π 2 — 14.10 1256 Undef. tan x approaches ∞ y y = tan x − π x approaches — 2 −π 2 − π 2 π 2 π 3π x 2 −2 period: π y y = cot x 2 −π − π 2 π 2 π 3π 2 x 2π period: π Core Concept Characteristics of y = tan x and y = cot x STUDY TIP The functions y = tan x and y = cot x have the following characteristics. π • The domain of y = tan x is all real numbers except odd multiples of —. 2 At these x-values, the graph has vertical asymptotes. π Odd multiples of — are 2 values such as these: • The domain of y = cot x is all real numbers except multiples of π. At these x-values, the graph has vertical asymptotes. π π ±1 — = ± — 2 2 π 3π ±3 — = ± — 2 2 π 5π ±5 — = ± — 2 2 • The range of each function is all real numbers. So, the functions do not have maximum or minimum values, and the graphs do not have an amplitude. ⋅ ⋅ ⋅ 498 Chapter 9 hsnb_alg2_pe_0905.indd 498 • The period of each graph is π. • The x-intercepts for y = tan x occur when x = 0, ±π, ±2π, ±3π, . . .. π 3π 5π 7π • The x-intercepts for y = cot x occur when x = ± —, ± —, ± —, ± —, . . .. 2 2 2 2 Trigonometric Ratios and Functions 2/5/15 1:51 PM Graphing Tangent and Cotangent Functions The graphs of y = a tan bx and y = a cot bx represent transformations of their parent functions. The value of a indicates a vertical stretch (a > 1) or a vertical shrink (0 < a < 1). The value of b indicates a horizontal stretch (0 < b < 1) or a horizontal shrink (b > 1) and changes the period of the graph. Core Concept Period and Vertical Asymptotes of y = a tan bx and y = a cot bx The period and vertical asymptotes of the graphs of y = a tan bx and y = a cot bx, where a and b are nonzero real numbers, are as follows. π • The period of the graph of each function is —. ∣b∣ π • The vertical asymptotes for y = a tan bx are at odd multiples of —. 2∣ b ∣ π • The vertical asymptotes for y = a cot bx are at multiples of —. ∣b∣ Each graph below shows five key x-values that you can use to sketch the graphs of y = a tan bx and y = a cot bx for a > 0 and b > 0. These are the x-intercept, the x-values where the asymptotes occur, and the x-values halfway between the x-intercept and the asymptotes. At each halfway point, the value of the function is either a or −a. y y a − π 2b a π 4b π 2b π 4b x y = a tan bx π 2b π b x y = a cot bx Graphing a Tangent Function Graph one period of g(x) = 2 tan 3x. Describe the graph of g as a transformation of the graph of f (x) = tan x. y SOLUTION 4 π − 6 π 12 −4 π 6 x The function is of the form g(x) = a tan bx where a = 2 and b = 3. So, the period is π π — = —. ∣b∣ 3 Intercept: (0, 0) π π π x = −— = −—, or x = −— 2(3) 6 2∣ b ∣ π π π Asymptotes: x = — = —, or x = —; 2∣ b ∣ 2(3) 6 Halfway points: π π , 2 ) = ( , 2 ); ( 4bπ , a ) = ( 4(3) 12 — — — π π , −2 ) = ( − , −2 ) ( −4bπ , −a ) = ( −4(3) 12 — — — The graph of g is a vertical stretch by a factor of 2 and a horizontal shrink by a factor of —13 of the graph of f. Section 9.5 hsnb_alg2_pe_0905.indd 499 Graphing Other Trigonometric Functions 499 2/5/15 1:51 PM Graphing a Cotangent Function Graph one period of g(x) = cot —12 x. Describe the graph of g as a transformation of the graph of f (x) = cot x. SOLUTION The function is of the form g(x) = a cot bx where a = 1 and b = —12 . So, the period is π π — = — = 2π. ∣ b ∣ —1 y 2 2 x π 2 −2 π 2π π π Intercept: —, 0 = —, 0 = (π, 0) 2b 2 —12 ( (() ) ) π π Asymptotes: x = 0; x = — = —, or x = 2π ∣ b ∣ —1 2 (() ) 3π π π π 3π 3π Halfway points: —, a = —, 1 = —, 1 ; —, −a = —, −1 = —, −1 4b 2 4b 2 1 1 4 —2 4 —2 ( ) ( )( (() ) ) ( ) The graph of g is a horizontal stretch by a factor of 2 of the graph of f. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. STUDY TIP 1 Because sec x = —, cos x sec x is undefined for x-values at which cos x = 0. The graph of y = sec x has vertical asymptotes at these x-values. You can use similar reasoning to understand the vertical asymptotes of the graph of y = csc x. 1 1. g(x) = tan 2x 2. g(x) = —3 cot x 4. g(x) = 5 tan πx 3. g(x) = 2 cot 4x Graphing Secant and Cosecant Functions The graphs of secant and cosecant functions are related to the graphs of the parent functions y = sec x and y = csc x, which are shown below. 3 y y 2 y = sec x π − 2 π 2 5π x 2 y = cos x −2π y = sin x −π π x y = csc x period: 2π Core Concept period: 2π Characteristics of y = sec x and y = csc x The functions y = sec x and y = csc x have the following characteristics. π • The domain of y = sec x is all real numbers except odd multiples of —. 2 At these x-values, the graph has vertical asymptotes. • The domain of y = csc x is all real numbers except multiples of π. At these x-values, the graph has vertical asymptotes. • The range of each function is y ≤ −1 and y ≥ 1. So, the graphs do not have an amplitude. • The period of each graph is 2π. 500 Chapter 9 hsnb_alg2_pe_0905.indd 500 Trigonometric Ratios and Functions 2/5/15 1:51 PM To graph y = a sec bx or y = a csc bx, first graph the function y = a cos bx or y = a sin bx, respectively. Then use the asymptotes and several points to sketch a graph of the function. Notice that the value of b represents a horizontal stretch or 2π 1 shrink by a factor of —, so the period of y = a sec bx and y = a csc bx is —. ∣b∣ b Graphing a Secant Function Graph one period of g(x) = 2 sec x. Describe the graph of g as a transformation of the graph of f (x) = sec x. SOLUTION Step 1 Graph the function y = 2 cos x. 2π The period is — = 2π. 1 y y = 2 sec x 3 y = 2 cos x Step 2 Graph asymptotes of g. Because the asymptotes of g occur when 2 cos x = 0, π π 3π graph x = −—, x = —, and x = —. 2 2 2 π 2 π x −3 Step 3 Plot points on g, such as (0, 2) and (π, −2). Then use the asymptotes to sketch the curve. The graph of g is a vertical stretch by a factor of 2 of the graph of f. Graphing a Cosecant Function LOOKING FOR A PATTERN In Examples 3 and 4, notice that the plotted points are on both graphs. Also, these points represent a local maximum on one graph and a local minimum on the other graph. 1 Graph one period of g(x) = — csc πx. Describe the graph of g as a transformation of 2 the graph of f (x) = csc x. SOLUTION 1 2π Step 1 Graph the function y = — sin πx. The period is — = 2. 2 π Step 2 Graph asymptotes of g. Because the 1 asymptotes of g occur when — sin πx = 0, 2 graph x = 0, x = 1, and x = 2. y 1 ( ) 1 1 Step 3 Plot points on g, such as —, — and 2 2 3 1 —, −— . Then use the asymptotes to 2 2 sketch the curve. ( ) 1 x 2 y = 1 sin π x 2 y = 1 csc π x 2 1 The graph of g is a vertical shrink by a factor of — and a horizontal shrink by 2 1 a factor of — of the graph of f. π Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. 5. g(x) = csc 3x Section 9.5 hsnb_alg2_pe_0905.indd 501 1 6. g(x) = —2 sec x 7. g(x) = 2 csc 2x 8. g(x) = 2 sec πx Graphing Other Trigonometric Functions 501 2/5/15 1:51 PM 9.5 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. WRITING Explain why the graphs of the tangent, cotangent, secant, and cosecant functions do not have an amplitude. 2. COMPLETE THE SENTENCE The _______ and _______ functions are undefined for x-values at which sin x = 0. 3. COMPLETE THE SENTENCE The period of the function y = sec x is _____, and the period of y = cot x is _____. 4. WRITING Explain how to graph a function of the form y = a sec bx. Monitoring Progress and Modeling with Mathematics In Exercises 5–12, graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. (See Examples 1 and 2.) 16. USING EQUATIONS Which of the following are asymptotes of the graph of y = 3 tan 4x? 5. g(x) = 2 tan x 6. g(x) = 3 tan x π A x=— ○ 8 π B x=— ○ 4 7. g(x) = cot 3x 8. g(x) = cot 2x C x=0 ○ 5π D x = −— ○ 8 1 1 9. g(x) = 3 cot —4 x 10. g(x) = 4 cot —2 x 1 In Exercises 17–24, graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. (See Examples 3 and 4.) 1 11. g(x) = —2 tan πx 12. g(x) = —3 tan 2πx 13. ERROR ANALYSIS Describe and correct the error in finding the period of the function y = cot 3x. ✗ 2π 2π Period: — = — ∣b∣ 3 18. g(x) = 2 csc x 19. g(x) = sec 4x 20. g(x) = sec 3x 1 2 π 2 graph to write a function of the form y = a tan bx. 25. − graph each function. y y = 4 sin 3x 502 Chapter 9 hsnb_alg2_pe_0905.indd 502 π x 2 27. − π 2 π x 2 −1 −4 π 6 π 2 28. y y 2 x x −4 1 π 2 4 π 2 y b. f (x) = 4 csc 3x y = 3 cos 2x π 4 26. y 12 15. ANALYZING RELATIONSHIPS Use the given graph to y π 4 24. g(x) = csc — x ATTENDING TO PRECISION In Exercises 25 –28, use the A vertical stretch by a factor of 5 and a horizontal shrink by a factor of —12. 4 22. g(x) = — sec 2πx 23. g(x) = csc — x in describing the transformation of f (x) = tan x represented by g(x) = 2 tan 5x. a. f (x) = 3 sec 2x 1 4 21. g(x) = — sec πx 14. ERROR ANALYSIS Describe and correct the error ✗ 17. g(x) = 3 csc x − 1 2 5 1 2 x π − 4 π 4 x Trigonometric Ratios and Functions 2/5/15 1:51 PM 40. f (x) = 4 csc x; vertical stretch by a factor of 2 and a USING STRUCTURE In Exercises 29–34, match the reflection in the x-axis equation with the correct graph. Explain your reasoning. 41. MULTIPLE REPRESENTATIONS Which function has 29. g(x) = 4 tan x 30. g(x) = 4 cot x 31. g(x) = 4 csc πx 32. g(x) = 4 sec πx 33. g(x) = sec 2x 34. g(x) = csc 2x a greater local maximum value? Which has a greater local minimum value? Explain. 1 A. f (x) = —4 csc πx B. y 4 A. B. y y 1 π 2 π − 2 x π x 2 42. ANALYZING RELATIONSHIPS Order the functions D. y from the least average rate of change to the greatest π π average rate of change over the interval −— < x < —. 4 4 y A. 4 4 π 2 E. −4 F. 2 π − 2 π x 2 π − 2 π x 2 y C. x D. y 2 π 4 y x 1 2 1 −1 B. y 2 π x y π − 4 x −8 −1 C. π 4 −4 4 y 2 2 x 1 π − 2 −4 π − 2 π x 2 π x 2 35. WRITING Explain why there is more than one tangent function whose graph passes through the origin and has asymptotes at x = −π and x = π. 36. USING EQUATIONS Graph one period of each function. Describe the transformation of the graph of its parent function. a. g(x) = sec x + 3 b. g(x) = csc x − 2 c. g(x) = cot(x − π) d. g(x) = −tan x 43. REASONING You are standing on a bridge 140 feet above the ground. You look down at a car traveling away from the underpass. The distance d (in feet) the car is from the base of the bridge can be modeled by d = 140 tan θ. Graph the function. Describe what happens to θ as d increases. θ 140 ft WRITING EQUATIONS In Exercises 37– 40, write a rule for g that represents the indicated transformation of the graph of f. π 37. f (x) = cot 2x; translation 3 units up and — units left 2 38. f (x) = 2 tan x; translation π units right, followed by a horizontal shrink by a factor of —13 39. f (x) = 5 sec (x − π); translation 2 units down, followed by a reflection in the x-axis Section 9.5 hsnb_alg2_pe_0905.indd 503 d 44. USING TOOLS You use a video camera to pan up the Statue of Liberty. The height h (in feet) of the part of the Statue of Liberty that can be seen through your video camera after time t (in seconds) can be modeled π by h = 100 tan — t. Graph the function using a 36 graphing calculator. What viewing window did you use? Explain. Graphing Other Trigonometric Functions 503 2/5/15 1:51 PM 45. MODELING WITH MATHEMATICS You are standing 48. HOW DO YOU SEE IT? Use the graph to answer 120 feet from the base of a 260-foot building. You watch your friend go down the side of the building in a glass elevator. each question. y your friend 2 d −3 −1 1 3 x 260 − d θ you 120 ft a. What is the period of the graph? Not drawn to scale b. What is the range of the function? a. Write an equation that gives the distance d (in feet) your friend is from the top of the building as a function of the angle of elevation θ. b. Graph the function found in part (a). Explain how the graph relates to this situation. c. Is the function of the form f (x) = a csc bx or f (x) = a sec bx? Explain. 49. ABSTRACT REASONING Rewrite a sec bx in terms of cos bx. Use your results to explain the relationship between the local maximums and minimums of the cosine and secant functions. 46. MODELING WITH MATHEMATICS You are standing 300 feet from the base of a 200-foot cliff. Your friend is rappelling down the cliff. a. Write an equation that gives the distance d (in feet) your friend is from the top of the cliff as a function of the angle of elevation θ. 50. THOUGHT PROVOKING A trigonometric equation that is true for all values of the variable for which both sides of the equation are defined is called a trigonometric identity. Use a graphing calculator to graph the function b. Graph the function found in part (a). c. Use a graphing calculator to determine the angle of elevation when your friend has rappelled halfway down the cliff. x 1 x y = — tan — + cot — . 2 2 2 ( ) Use your graph to write a trigonometric identity involving this function. Explain your reasoning. 47. MAKING AN ARGUMENT Your friend states that it 51. CRITICAL THINKING Find a tangent function whose is not possible to write a cosecant function that has the same graph as y = sec x. Is your friend correct? Explain your reasoning. graph intersects the graph of y = 2 + 2 sin x only at minimum points of the sine function. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Write a cubic function whose graph passes through the given points. (Section 4.9) 52. (−1, 0), (1, 0), (3, 0), (0, 3) 53. (−2, 0), (1, 0), (3, 0), (0, −6) 54. (−1, 0), (2, 0), (3, 0), (1, −2) 55. (−3, 0), (−1, 0), (3, 0), (−2, 1) Find the amplitude and period of the graph of the function. (Section 9.4) 56. 57. y 58. y 6 2 5 π −5 504 Chapter 9 hsnb_alg2_pe_0905.indd 504 2π π 2 x y π 2π x 6π x −2 −6 Trigonometric Ratios and Functions 2/5/15 1:51 PM 9.6 Modeling with Trigonometric Functions Essential Question What are the characteristics of the real-life problems that can be modeled by trigonometric functions? Modeling Electric Currents MODELING WITH MATHEMATICS Work with a partner. Find a sine function that models the electric current shown in each oscilloscope screen. State the amplitude and period of the graph. a. To be proficient in math, you need to apply the mathematics you know to solve problems arising in everyday life. 15 10 10 5 5 0 0 -5 -5 -10 -10 -15 -15 1 2 3 4 5 6 7 8 9 -20 0 10 d. 20 15 10 10 5 5 0 0 -5 -5 -10 -10 -15 -15 1 2 3 4 5 6 7 8 9 -20 0 10 f. 20 15 10 10 5 5 0 0 -5 -5 -10 -10 -15 -15 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 20 15 -20 0 1 20 15 -20 0 e. 20 15 -20 0 c. b. 20 -20 0 Communicate Your Answer 2. What are the characteristics of the real-life problems that can be modeled by trigonometric functions? 3. Use the Internet or some other reference to find examples of real-life situations that can be modeled by trigonometric functions. Section 9.6 hsnb_alg2_pe_0906.indd 505 Modeling with Trigonometric Functions 505 2/5/15 1:52 PM What You Will Learn 9.6 Lesson Interpret and use frequency. Write trigonometric functions. Core Vocabul Vocabulary larry Use technology to find trigonometric models. frequency, p. 506 sinusoid, p. 507 Frequency Previous amplitude period midline The periodic nature of trigonometric functions makes them useful for modeling oscillating motions or repeating patterns that occur in real life. Some examples are sound waves, the motion of a pendulum, and seasons of the year. In such applications, the reciprocal of the period is called the frequency, which gives the number of cycles per unit of time. Using Frequency A sound consisting of a single frequency is called a pure tone. An audiometer produces pure tones to test a person’s auditory functions. An audiometer produces a pure tone with a frequency f of 2000 hertz (cycles per second). The maximum pressure P produced from the pure tone is 2 millipascals. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds). SOLUTION Step 1 Find the values of a and b in the model P = a sin bt. The maximum pressure is 2, so a = 2. Use the frequency f to find b. 1 frequency = — period Write relationship involving frequency and period. b 2000 = — 2π Substitute. 4000π = b Multiply each side by 2π. The pressure P as a function of time t is given by P = 2 sin 4000π t. Step 2 Graph the model. The amplitude is a = 2 and the period is P 2 1 f 1 2000 — = —. 1 8000 −2 t The key points are: ( ⋅ ) ( )( 1 1 1 1 Intercepts: (0, 0); — —, 0 = —, 0 ; —, 0 2 2000 4000 2000 ) (⋅ ) ( ) 3 1 3 , −2 ) = ( , −2 ) Minimum: ( ⋅ 4 2000 8000 1 1 1 Maximum: — —, 2 = —, 2 4 2000 8000 — — — The graph of P = 2 sin 4000π t is shown at the left. 506 Chapter 9 hsnb_alg2_pe_0906.indd 506 Trigonometric Ratios and Functions 2/5/15 1:52 PM Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. WHAT IF? In Example 1, how would the function change when the audiometer produced a pure tone with a frequency of 1000 hertz? Writing Trigonometric Functions Graphs of sine and cosine functions are called sinusoids. One method to write a sine or cosine function that models a sinusoid is to find the values of a, b, h, and k for y = a sin b(x − h) + k or y = a cos b(x − h) + k 2π where ∣ a ∣ is the amplitude, — is the period (b > 0), h is the horizontal shift, and k is b the vertical shift. Writing a Trigonometric Function Write a function for the sinusoid shown. y 5 (π8 , 5( 3 x π ( 38π , −1( SOLUTION Step 1 Find the maximum and minimum values. From the graph, the maximum value is 5 and the minimum value is −1. Step 2 Identify the vertical shift, k. The value of k is the mean of the maximum and minimum values. (maximum value) + (minimum value) 5 + (−1) 4 k = ———— = — = — = 2 2 2 2 STUDY TIP Because the graph repeats π every — units, the period 2 π is —. 2 Step 3 Decide whether the graph should be modeled by a sine or cosine function. Because the graph crosses the midline y = 2 on the y-axis, the graph is a sine curve with no horizontal shift. So, h = 0. Step 4 Find the amplitude and period. The period is Check π 2 2π b —=— 6 b = 4. The amplitude is (maximum value) − (minimum value) 5 − (−1) 6 ∣ a ∣ = ———— = — = — = 3. 2 π − 2 2π −2 2 The graph is not a reflection, so a > 0. Therefore, a = 3. The function is y = 3 sin 4x + 2. Check this by graphing the function on a graphing calculator. Section 9.6 hsnb_alg2_pe_0906.indd 507 2 Modeling with Trigonometric Functions 507 2/5/15 1:52 PM Modeling Circular Motion Two people swing jump ropes, as shown in the diagram. The highest point of the middle of each rope is 75 inches above the ground, and the lowest point is 3 inches. The rope makes 2 revolutions per second. Write a model for the height h (in inches) of a rope as a function of the time t (in seconds) given that the rope is at its lowest point when t = 0. 75 in. above ground 3 in. above ground Not drawn to scale SOLUTION A rope oscillates between 3 inches and 75 inches above the ground. So, a sine or cosine function may be an appropriate model for the height over time. Step 1 Identify the maximum and minimum values. The maximum height of a rope is 75 inches. The minimum height is 3 inches. Step 2 Identify the vertical shift, k. (maximum value) + (minimum value) 75 + 3 k = ———— = — = 39 2 2 Check Use the table feature of a graphing calculator to check your model. X Y1 .25 .5 .75 1 1.25 1.5 3 75 3 75 3 75 3 Step 3 Decide whether the height should be modeled by a sine or cosine function. When t = 0, the height is at its minimum. So, use a cosine function whose graph is a reflection in the x-axis with no horizontal shift (h = 0). Step 4 Find the amplitude and period. (maximum value) − (minimum value) 75 − 3 The amplitude is ∣ a ∣ = ———— = — = 36. 2 2 2 revolutions Because the graph is a reflection in the x-axis, a < 0. So, a = −36. Because a rope is rotating at a rate of 2 revolutions per second, one revolution is 2π completed in 0.5 second. So, the period is — = 0.5, and b = 4π. b X=0 A model for the height of a rope is h(t) = −36 cos 4πt + 39. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write a function for the sinusoid. 2. 3 3. y (0, 2) −1 −3 (π3 , −2( (12 , 1( y 1 2π 3 x 1 2 3 2 5 2 x (32 , −3( 4. WHAT IF? Describe how the model in Example 3 changes when the lowest point of a rope is 5 inches above the ground and the highest point is 70 inches above the ground. 508 Chapter 9 hsnb_alg2_pe_0906.indd 508 Trigonometric Ratios and Functions 2/5/15 1:52 PM Using Technology to Find Trigonometric Models Another way to model sinusoids is to use a graphing calculator that has a sinusoidal regression feature. Using Sinusoidal Regression T table shows the numbers N of hours of daylight in Denver, Colorado, on the The 115th day of each month, where t = 1 represents January. Write a model that gives N aas a function of t and interpret the period of its graph. t 1 2 3 4 5 6 N 9.68 10.75 11.93 13.27 14.38 14.98 t 7 8 9 10 11 12 N 14.70 13.73 12.45 11.17 9.98 9.38 SOLUTION S Step 1 Enter the data in a graphing S calculator. L1 L2 1 2 3 4 5 6 7 L3 9.68 10.75 11.93 13.27 14.38 14.98 14.7 Step 2 20 1 ------ 0 L1(1)=1 Step 3 The scatter plot appears sinusoidal. So, perform a sinusoidal regression. STUDY TIP Notice that the sinusoidal regression feature finds a model of the form y = a sin(bx + c) + d. This 2π function has a period of — b because it can be written c as y = a sin b x + — + d. b ( ) Make a scatter plot. 13 0 Step 4 Graph the data and the model in the same viewing window. 20 SinReg y=a*sin(bx+c)+d a=2.764734198 b=.5111635715 c=-1.591149599 d=12.13293913 0 13 0 The model appears to be a good fit. So, a model for the data is 2π N = 2.76 sin(0.511t − 1.59) + 12.1. The period, — ≈ 12, makes sense 0.511 because there are 12 months in a year and you would expect this pattern to continue in following years. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. The table shows the average daily temperature T (in degrees Fahrenheit) for a city each month, where m = 1 represents January. Write a model that gives T as a function of m and interpret the period of its graph. m 1 2 3 4 5 6 7 8 9 10 11 12 T 29 32 39 48 59 68 74 72 65 54 45 35 Section 9.6 hsnb_alg2_pe_0906.indd 509 Modeling with Trigonometric Functions 509 2/5/15 1:52 PM 9.6 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Graphs of sine and cosine functions are called __________. 2. WRITING Describe how to find the frequency of the function whose graph is shown. y 0.1 1 12 x Monitoring Progress and Modeling with Mathematics In Exercises 3 –10, find the frequency of the function. 3. y = sin x 4. y = sin 3x 5. y = cos 4x + 2 6. y = −cos 2x 7. y = sin 3πx 8. y = cos — In Exercises 13–16, write a function for the sinusoid. (See Example 2.) 13. 1 2 9. y = — cos 0.75x − 8 y (π4 , 3) 2 x πx 4 − 3π 4 π − 4 π 4 3π 4 10. y = 3 sin 0.2x + 6 11. MODELING WITH MATHEMATICS The lowest 14. frequency of sounds that can be heard by humans is 20 hertz. The maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds). (See Example 1.) 12. MODELING WITH MATHEMATICS A middle-A tuning fork vibrates with a frequency f of 440 hertz (cycles per second). You strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds). 5π 4 7π 4 ( 34π , −3) y 6 π − 2 (0, 5) π 2 −2 (π4 , −5) −6 15. y x (2, 2) 2 2 4 x 6 (0, −2) 16. y −1 1 ( 32 , −1( 4 x −2 ( 12 , −3( 510 Chapter 9 hsnb_alg2_pe_0906.indd 510 Trigonometric Ratios and Functions 2/5/15 1:52 PM 17. ERROR ANALYSIS Describe and correct the error in finding the amplitude of a sinusoid with a maximum point at (2, 10) and a minimum point at (4, −6). ✗ (maximum value) + (minimum value) ∣ a ∣ = ——— 2 USING TOOLS In Exercises 21 and 22, the time t is measured in months, where t = 1 represents January. Write a model that gives the average monthly high temperature D as a function of t and interpret the period of the graph. (See Example 4.) 21. 10 − 6 =— 2 =2 18. ERROR ANALYSIS Describe and correct the error in finding the vertical shift of a sinusoid with a maximum point at (3, −2) and a minimum point at (7, −8). ✗ (maximum value) + (minimum value) k = ——— 2 7+3 =— 2 =5 22. Air Temperatures in Apple Valley, CA t 1 2 3 4 5 6 D 60 63 69 75 85 94 t 7 8 9 10 11 12 D 99 99 93 81 69 60 Water Temperatures at Miami Beach, FL t 1 2 3 4 5 6 D 71 73 75 78 81 85 t 7 8 9 10 11 12 D 86 85 84 81 76 73 23. MODELING WITH MATHEMATICS A circuit has an 19. MODELING WITH MATHEMATICS One of the largest sewing machines in the world has a flywheel (which turns as the machine sews) that is 5 feet in diameter. The highest point of the handle at the edge of the flywheel is 9 feet above the ground, and the lowest point is 4 feet. The wheel makes a complete turn every 2 seconds. Write a model for the height h (in feet) of the handle as a function of the time t (in seconds) given that the handle is at its lowest point when t = 0. (See Example 3.) 20. MODELING WITH MATHEMATICS The Great Laxey Wheel, located on the Isle of Man, is the largest working water wheel in the world. The highest point of a bucket on the wheel is 70.5 feet above the viewing platform, and the lowest point is 2 feet below the viewing platform. The wheel makes a complete turn every 24 seconds. Write a model for the height h (in feet) of the bucket as a function of time t (in seconds) given that the bucket is at its lowest point when t = 0. alternating voltage of 100 volts that peaks every 0.5 second. Write a sinusoidal model for the voltage V as a function of the time t (in seconds). ( 18 , 100( V 100 t 1 8 ( 38 , −100( 24. MULTIPLE REPRESENTATIONS The graph shows the average daily temperature of Lexington, Kentucky. The average daily temperature of Louisville, π Kentucky, is modeled by y = −22 cos —t + 57, 6 where y is the temperature (in degrees Fahrenheit) and t is the number of months since January 1. Which city has the greater average daily temperature? Explain. Temperature (F°) Daily Temperature in Lexington T 80 (6, 76) 40 (0, 33) 0 0 2 4 6 8 10 t Months since January 1 Section 9.6 hsnb_alg2_pe_0906.indd 511 Modeling with Trigonometric Functions 511 2/5/15 1:52 PM 25. USING TOOLS The table shows the numbers of 28. HOW DO YOU SEE IT? What is the frequency of the employees N (in thousands) at a sporting goods company each year for 11 years. The time t is measured in years, with t = 1 representing the first year. t 1 2 3 4 5 6 N 20.8 22.7 24.6 23.2 20 17.5 t 7 8 9 10 11 12 N 16.7 17.8 21 22 24.1 function whose graph is shown? Explain. y 0.5 x 1 8 13 8 17 8 21 8 π π has a minimum at —, 3 and a maximum at —, 8 . 2 4 Write a sine function and a cosine function for the sinusoid. Use a graphing calculator to verify that your answers are correct. ( ) b. Predict the number of employees at the company in the 12th year. 26. THOUGHT PROVOKING The figure shows a tangent ( ) 30. MAKING AN ARGUMENT Your friend claims that a line drawn to the graph of the function y = sin x. At several points on the graph, draw a tangent line to the graph and estimate its slope. Then plot the points (x, m), where m is the slope of the tangent line. What can you conclude? function with a frequency of 2 has a greater period than a function with a frequency of —12. Is your friend correct? Explain your reasoning. 31. PROBLEM SOLVING The low tide at a port is 3.5 feet y and occurs at midnight. After 6 hours, the port is at high tide, which is 16.5 feet. 1 −π 9 8 29. USING STRUCTURE During one cycle, a sinusoid a. Use sinusoidal regression to find a model that gives N as a function of t. −2π 5 8 π 2π x The slope of the tangent line at (0, 0) is 1. high h tid ide: 16.5 5 ft low lo w tiide e: 3.5 5 ft ft 27. REASONING Determine whether you would use a sine or cosine function to model each sinusoid with the y-intercept described. Explain your reasoning. a. Write a sinusoidal model that gives the tide depth d (in feet) as a function of the time t (in hours). Let t = 0 represent midnight. a. The y-intercept occurs at the maximum value of the function. b. Find all the times when low and high tides occur in a 24-hour period. b. The y-intercept occurs at the minimum value of the function. c. Explain how the graph of the function you wrote in part (a) is related to a graph that shows the tide depth d at the port t hours after 3:00 a.m. c. The y-intercept occurs halfway between the maximum and minimum values of the function. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Simplify the expression. (Section 5.2) 3 17 32. — — √2 33. — — √6 − 2 8 13 34. — — 35. — — — √3 + √11 38. log3 5x3 39. ln — √10 + 3 Expand the logarithmic expression. (Section 6.5) x 7 36. log8 — 512 Chapter 9 hsnb_alg2_pe_0906.indd 512 37. ln 2x 4x6 y Trigonometric Ratios and Functions 2/5/15 1:52 PM 9.7 Using Trigonometric Identities Essential Question How can you verify a trigonometric identity? Writing a Trigonometric Identity Work with a partner. In the figure, the point (x, y) is on a circle of radius c with center at the origin. y (x, y) a. Write an equation that relates a, b, and c. b. Write expressions for the sine and cosine ratios of angle θ. c c. Use the results from parts (a) and (b) to find the sum of sin2θ and cos2θ. What do you observe? θ b a x d. Complete the table to verify that the identity you wrote in part (c) is valid for angles (of your choice) in each of the four quadrants. θ sin2 θ cos2 θ sin2 θ + cos2 θ QI QII QIII QIV Writing Other Trigonometric Identities REASONING ABSTRACTLY To be proficient in math, you need to know and flexibly use different properties of operations and objects. Work with a partner. The trigonometric identity you derived in Exploration 1 is called a Pythagorean identity. There are two other Pythagorean identities. To derive them, recall the four relationships: sin θ tan θ = — cos θ cos θ cot θ = — sin θ 1 sec θ = — cos θ 1 csc θ = — sin θ a. Divide each side of the Pythagorean identity you derived in Exploration 1 by cos2θ and simplify. What do you observe? b. Divide each side of the Pythagorean identity you derived in Exploration 1 by sin2θ and simplify. What do you observe? Communicate Your Answer 3. How can you verify a trigonometric identity? 4. Is sin θ = cos θ a trigonometric identity? Explain your reasoning. 5. Give some examples of trigonometric identities that are different than those in Explorations 1 and 2. Section 9.7 hsnb_alg2_pe_0907.indd 513 Using Trigonometric Identities 513 2/5/15 1:53 PM 9.7 Lesson What You Will Learn Use trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. Core Vocabul Vocabulary larry Verify trigonometric identities. trigonometric identity, p. 514 Previous unit circle Using Trigonometric Identities Recall that when an angle θ is in standard position with its terminal side intersecting the unit circle at (x, y), then x = cos θ and y = sin θ. Because (x, y) is on a circle centered at the origin with radius 1, it follows that STUDY TIP Note that sin2 θ represents (sin θ)2 and cos2 θ represents (cos θ)2. y r=1 (cos θ, sin θ) = (x, y) θ x x2 + y2 = 1 and cos2 θ + sin2 θ = 1. The equation cos2 θ + sin2 θ = 1 is true for any value of θ. A trigonometric equation that is true for all values of the variable for which both sides of the equation are defined is called a trigonometric identity. In Section 9.1, you used reciprocal identities to find the values of the cosecant, secant, and cotangent functions. These and other fundamental trigonometric identities are listed below. Core Concept Fundamental Trigonometric Identities Reciprocal Identities 1 csc θ = — sin θ 1 sec θ = — cos θ 1 cot θ = — tan θ Tangent and Cotangent Identities sin θ tan θ = — cos θ cos θ cot θ = — sin θ Pythagorean Identities sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ π cos — − θ = sin θ 2 π tan — − θ = cot θ 2 cos(−θ) = cos θ tan(−θ) = −tan θ Cofunction Identities π sin — − θ = cos θ 2 ( ) ( ) ( ) Negative Angle Identities sin(−θ) = −sin θ In this section, you will use trigonometric identities to do the following. • Evaluate trigonometric functions. • Simplify trigonometric expressions. • Verify other trigonometric identities. 514 Chapter 9 hsnb_alg2_pe_0907.indd 514 Trigonometric Ratios and Functions 2/5/15 1:54 PM Finding Trigonometric Values 4 π Given that sin θ = — and — < θ < π, find the values of the other five trigonometric 5 2 functions of θ. SOLUTION Step 1 Find cos θ. sin2 θ + cos2 θ = 1 Write Pythagorean identity. 2 ( 45 ) + cos θ = 1 4 Substitute — for sin θ. 5 2 — 4 2 42 cos2 θ = 1 − — Subtract — from each side. 5 5 9 cos2 θ = — Simplify. 25 3 cos θ = ± — Take square root of each side. 5 3 cos θ = −— Because θ is in Quadrant II, cos θ is negative. 5 Step 2 Find the values of the other four trigonometric functions of θ using the values of sin θ and cos θ. 4 3 −— — 5 5 4 3 sin θ cos θ cot θ = — = — = −— tan θ = — = — = −— cos θ 3 sin θ 4 3 4 −— — 5 5 () () 1 5 1 csc θ = — = — = — sin θ 4 4 — 5 1 5 1 sec θ = — = — = −— cos θ 3 3 −— 5 Simplifying Trigonometric Expressions π Simplify (a) tan — − θ sin θ and (b) sec θ tan2 θ + sec θ. 2 ( ) SOLUTION π a. tan — − θ sin θ = cot θ sin θ 2 cos θ = — (sin θ) sin θ = cos θ ( ) Cofunction identity ( ) Cotangent identity Simplify. b. sec θ tan2 θ + sec θ = sec θ(sec2 θ − 1) + sec θ Pythagorean identity = sec3 θ − sec θ + sec θ Distributive Property = sec3 θ Simplify. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1 6 trigonometric functions of θ. π 2 1. Given that cos θ = — and 0 < θ < —, find the values of the other five Simplify the expression. 2. sin x cot x sec x 3. cos θ − cos θ sin2 θ Section 9.7 hsnb_alg2_pe_0907.indd 515 tan x csc x sec x 4. — Using Trigonometric Identities 515 2/5/15 1:54 PM Verifying Trigonometric Identities You can use the fundamental identities from this chapter to verify new trigonometric identities. When verifying an identity, begin with the expression on one side. Use algebra and trigonometric properties to manipulate the expression until it is identical to the other side. Verifying a Trigonometric Identity sec2 θ − 1 Verify the identity — = sin2 θ. sec2 θ SOLUTION sec2 θ − 1 sec θ sec2 θ sec θ 1 sec θ =— −— — 2 2 2 Write as separate fractions. 2 ( ) 1 =1− — sec θ Simplify. = 1 − cos2 θ Reciprocal identity = sin2 θ Pythagorean identity Notice that verifying an identity is not the same as solving an equation. When verifying an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. So, you cannot use any properties of equality, such as adding the same quantity to each side of the equation. Verifying a Trigonometric Identity cos x Verify the identity sec x + tan x = —. 1 − sin x SOLUTION LOOKING FOR STRUCTURE To verify the identity, you must introduce 1 − sin x into the denominator. Multiply the numerator and the denominator by 1 − sin x so you get an equivalent expression. 1 sec x + tan x = — + tan x cos x Reciprocal identity sin x 1 =—+— cos x cos x Tangent identity 1 + sin x =— cos x Add fractions. 1 + sin x 1 − sin x =— — cos x 1 − sin x 1 − sin x Multiply by —. 1 − sin x 1 − sin2 x = —— cos x(1 − sin x) Simplify numerator. cos2 x = —— cos x(1 − sin x) Pythagorean identity cos x =— 1 − sin x Simplify. ⋅ Monitoring Progress Help in English and Spanish at BigIdeasMath.com Verify the identity. 516 Chapter 9 hsnb_alg2_pe_0907.indd 516 5. cot(−θ) = −cot θ 6. csc2 x(1 − sin2 x) = cot2 x 7. cos x csc x tan x = 1 8. (tan2 x + 1)(cos2 x − 1) = −tan2 x Trigonometric Ratios and Functions 2/5/15 1:54 PM 9.7 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. WRITING Describe the difference between a trigonometric identity and a trigonometric equation. 2. WRITING Explain how to use trigonometric identities to determine whether sec(−θ) = sec θ or sec(−θ) = −sec θ. Monitoring Progress and Modeling with Mathematics In Exercises 3 –10, find the values of the other five trigonometric functions of θ. (See Example 1.) π 2 1 3 3. sin θ = —, 0 < θ < — ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in simplifying the expression. 21. ✗ 22. ✗ 3π 2 7 10 4. sin θ = −—, π < θ < — 3 π 7 2 5. tan θ = −—, — < θ < π 2 π 5 2 6. cot θ = −—, — < θ < π 3π 2 5 6 1 − sin2 θ = 1 − (1 + cos2 θ ) = 1 − 1 − cos2 θ = −cos2 θ cos x 1 tan x csc x = — — sin x sin x cos x =— sin2 x ⋅ 7. cos θ = −—, π < θ < — 9 3π 4 2 8. sec θ = —, — < θ < 2π In Exercises 23–30, verify the identity. (See Examples 3 and 4.) 3π 2 9. cot θ = −3, — < θ < 2π 23. sin x csc x = 1 3π 2 5 3 24. tan θ csc θ cos θ = 1 ( π2 ) π sin( − x ) tan x = sin x 2 π cos( − θ ) + 1 2 = 1 28. 10. csc θ = −—, π < θ < — 25. cos — − x cot x = cos x In Exercises 11–20, simplify the expression. (See Example 2.) 26. 12. cos θ (1 + tan2 θ) 11. sin x cot x sin(−θ) cos(−θ) cos2 x cot x 13. — 14. — 2 π cos — − x 2 15. — csc x ( ) ( π2 ) 16. sin — − θ sec θ csc2 x − cot2 x sin(−x) cot x cos2 x tan2(−x) − 1 cos x 17. —— 18. —— 2 π cos — − θ 2 19. — + cos2 θ csc θ π sec x sin x + cos — − x 2 20. ——— 1 + sec x ( ) ( ) — — 27. —— 1 − sin(−θ) 1 + cos x sin x sin x 1 + cos x 29. — + — = 2 csc x sin x 1 − cos(−x) 30. —— = csc x + cot x 31. USING STRUCTURE A function f is odd when f (−x) = −f(x). A function f is even when f (−x) = f (x). Which of the six trigonometric functions are odd? Which are even? Justify your answers using identities and graphs. 32. ANALYZING RELATIONSHIPS As the value of cos θ increases, what happens to the value of sec θ? Explain your reasoning. Section 9.7 hsnb_alg2_pe_0907.indd 517 sin2(−x) tan x = cos2 x — 2 Using Trigonometric Identities 517 2/5/15 1:54 PM 33. MAKING AN ARGUMENT Your friend simplifies 37. DRAWING CONCLUSIONS Static friction is the amount an expression and obtains sec x tan x − sin x. You simplify the same expression and obtain sin x tan2 x. Are your answers equivalent? Justify your answer. of force necessary to keep a stationary object on a flat surface from moving. Suppose a book weighing W pounds is lying on a ramp inclined at an angle θ. The coefficient of static friction u for the book can be found using the equation uW cos θ = W sin θ. 34. HOW DO YOU SEE IT? The figure shows the unit circle and the angle θ. a. Solve the equation for u and simplify the result. a. Is sin θ positive or negative? cos θ? tan θ? b. Use the equation from part (a) to determine what happens to the value of u as the angle θ increases from 0° to 90°. b. In what quadrant does the terminal side of −θ lie? c. Is sin(−θ) positive or negative? cos(−θ)? tan(−θ)? 38. PROBLEM SOLVING When light traveling in a medium (such as air) strikes the surface of a second medium (such as water) at an angle θ1, the light begins to travel at a different angle θ2. This change of direction is defined by Snell’s law, n1 sin θ1 = n2 sin θ2, where n1 and n2 are the indices of refraction for the two mediums. Snell’s law can be derived from the equation y (x, y) θ x n1 n2 √cot2 θ1 + 1 √cot2 θ2 + 1 —— — = —— —. 35. MODELING WITH MATHEMATICS A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the Sun above the horizon is θ can be modeled by the equation below. Show that the equation below is equivalent to s = h cot θ. h sin(90° − θ) s =—— sin θ h s air: n1 θ1 water: n2 θ2 a. Simplify the equation to derive Snell’s law. b. What is the value of n1 when θ1 = 55°, θ2 = 35°, and n2 = 2? c. If θ1 = θ2, then what must be true about the values of n1 and n2? Explain when this situation would occur. 39. WRITING Explain how transformations of the graph of the parent function f (x) = sin x support the π cofunction identity sin — − θ = cos θ. 2 θ ) ( 40. USING STRUCTURE Verify each identity. a. ln∣ sec θ ∣ = −ln∣ cos θ ∣ 36. THOUGHT PROVOKING Explain how you can use a trigonometric identity to find all the values of x for which sin x = cos x. Maintaining Mathematical Proficiency b. ln∣ tan θ ∣ = ln∣ sin θ ∣ − ln∣ cos θ ∣ Reviewing what you learned in previous grades and lessons Find the value of x for the right triangle. (Section 9.1) 41. 42. 43. 13 11 x 7 30° 45° x 518 Chapter 9 hsnb_alg2_pe_0907.indd 518 60° x Trigonometric Ratios and Functions 2/5/15 1:54 PM 9.8 Using Sum and Difference Formulas Essential Question How can you evaluate trigonometric functions of the sum or difference of two angles? Deriving a Difference Formula Work with a partner. a. Explain why the two triangles shown are congruent. y (cos a, sin a) y d (cos(a − b), sin(a − b)) (cos b, sin b) 1 a d 1 a−b b x CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. (1, 0) x b. Use the Distance Formula to write an expression for d in the first unit circle. c. Use the Distance Formula to write an expression for d in the second unit circle. d. Write an equation that relates the expressions in parts (b) and (c). Then simplify this equation to obtain a formula for cos(a − b). Deriving a Sum Formula Work with a partner. Use the difference formula you derived in Exploration 1 to write a formula for cos(a + b) in terms of sine and cosine of a and b. Hint: Use the fact that cos(a + b) = cos[a − (−b)]. Deriving Difference and Sum Formulas Work with a partner. Use the formulas you derived in Explorations 1 and 2 to write formulas for sin(a − b) and sin(a + b) in terms of sine and cosine of a and b. Hint: Use the cofunction identities π π sin — − a = cos a and cos — − a = sin a 2 2 ( ) ( ) and the fact that π cos — − a + b = sin(a − b) and sin(a + b) = sin[a − (−b)]. 2 [( ) ] Communicate Your Answer 4. How can you evaluate trigonometric functions of the sum or difference of two angles? 5. a. Find the exact values of sin 75° and cos 75° using sum formulas. Explain your reasoning. b. Find the exact values of sin 75° and cos 75° using difference formulas. Compare your answers to those in part (a). Section 9.8 hsnb_alg2_pe_0908.indd 519 Using Sum and Difference Formulas 519 2/5/15 1:54 PM 9.8 Lesson What You Will Learn Use sum and difference formulas to evaluate and simplify trigonometric expressions. Core Vocabul Vocabulary larry Use sum and difference formulas to solve trigonometric equations and rewrite real-life formulas. Previous ratio Using Sum and Difference Formulas In this lesson, you will study formulas that allow you to evaluate trigonometric functions of the sum or difference of two angles. Core Concept Sum and Difference Formulas Difference Formulas Sum Formulas sin(a + b) = sin a cos b + cos a sin b sin(a − b) = sin a cos b − cos a sin b cos(a + b) = cos a cos b − sin a sin b cos(a − b) = cos a cos b + sin a sin b tan a + tan b tan(a + b) = —— 1 − tan a tan b tan a − tan b tan(a − b) = —— 1 + tan a tan b In general, sin(a + b) ≠ sin a + sin b. Similar statements can be made for the other trigonometric functions of sums and differences. Evaluating Trigonometric Expressions 7π Find the exact value of (a) sin 15° and (b) tan —. 12 SOLUTION a. sin 15° = sin(60° − 45°) Check Substitute 60° − 45° for 15°. = sin 60° cos 45° − cos 60° sin 45° sin(15˚) .2588190451 — — ( ) ( ) √3 √2 1 √2 =— — −— — 2 2 2 2 ( (6)- (2))/4 .2588190451 — Difference formula for sine — Evaluate. — √6 − √2 =— 4 Simplify. — — √6 − √2 The exact value of sin 15° is —. Check this with a calculator. 4 π π 7π b. tan — = tan — + — 12 3 4 π π tan — + tan — 3 4 = —— π π 1 − tan — tan — 3 4 ( Check tan(7π/12) -3.732050808 -2- (3) -3.732050808 ) π π 7π Substitute — + — for —. 12 3 4 Sum formula for tangent — √3 + 1 =— — 1 − √3 1 ⋅ Evaluate. = −2 − √ 3 Simplify. — — 7π The exact value of tan — is −2 − √ 3 . Check this with a calculator. 12 520 Chapter 9 hsnb_alg2_pe_0908.indd 520 Trigonometric Ratios and Functions 2/5/15 1:54 PM Using a Difference Formula ANOTHER WAY You can also use a Pythagorean identity and quadrant signs to find sin a and cos b. 4 5 3π Find cos(a − b) given that cos a = −— with π < a < — and sin b = — with 5 2 13 π 0 < b < —. 2 SOLUTION Step 1 Find sin a and cos b. 4 Because cos a = −— and a is in 5 3 Quadrant III, sin a = −—, as 5 shown in the figure. 5 Because sin b = — and b is in 13 12 Quadrant I, cos b = —, as shown 13 in the figure. y y 4 52 − 42 = 3 13 b a 5 x x 132 − 52 = 12 5 Step 2 Use the difference formula for cosine to find cos(a − b). cos(a − b) = cos a cos b + sin a sin b ( ) ( )( ) Difference formula for cosine 4 12 3 5 = −— — + −— — 5 13 5 13 Evaluate. 63 = −— 65 Simplify. 63 The value of cos(a − b) is −—. 65 Simplifying an Expression Simplify the expression cos(x + π). SOLUTION cos(x + π) = cos x cos π − sin x sin π Sum formula for cosine = (cos x)(−1) − (sin x)(0) Evaluate. = −cos x Simplify. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the exact value of the expression. π 12 π 24 8 5. Find sin(a − b) given that sin a = — with 0 < a < — and cos b = −— 17 2 25 3π with π < b < —. 2 1. sin 105° 5π 12 2. cos 15° 3. tan — 4. cos — Simplify the expression. 6. sin(x + π) 7. cos(x − 2π) Section 9.8 hsnb_alg2_pe_0908.indd 521 8. tan(x − π) Using Sum and Difference Formulas 521 2/5/15 1:54 PM Solving Equations and Rewriting Formulas Solving a Trigonometric Equation π π Solve sin x + — + sin x − — = 1 for 0 ≤ x < 2π. 3 3 ( ) ( ) SOLUTION ANOTHER WAY You can also solve the equation by using a graphing calculator. First, graph each side of the original equation. Then use the intersect feature to find the x-value(s) where the expressions are equal. π π sin x + — + sin x − — = 1 3 3 π π π π sin x cos — + cos x sin — + sin x cos — − cos x sin — = 1 3 3 — 3 3 — √3 √3 1 1 —sin x + —cos x + —sin x − —cos x = 1 2 2 2 2 sin x = 1 ( ) ( ) Write equation. Use formulas. Evaluate. Simplify. π In the interval 0 ≤ x < 2π, the solution is x = —. 2 Rewriting a Real-Life Formula air α θ t h lig prism The index of refraction of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. A triangular prism, like the one shown, can be used to measure the index of refraction using the formula θ α sin — + — 2 2 n = —. θ sin — 2 ( ) — √3 1 θ For α = 60°, show that the formula can be rewritten as n = — + — cot —. 2 2 2 SOLUTION ) θ sin — + 30° 2 n = —— θ sin — 2 θ θ sin — cos 30° + cos — sin 30° 2 2 = ——— θ sin — 2 — θ √3 θ 1 sin — — + cos — — 2 2 2 2 = ——— θ sin — 2 — √3 θ 1 θ —sin — — cos — 2 2 2 2 =—+— θ θ sin — sin — 2 2 — √3 1 θ = — + — cot — 2 2 2 ( ( )( ) ( )( ) Monitoring Progress ( π4 ) ( α 60° Write formula with — = — = 30°. 2 2 Sum formula for sine Evaluate. Write as separate fractions. Simplify. Help in English and Spanish at BigIdeasMath.com π 4 ) 9. Solve sin — − x − sin x + — = 1 for 0 ≤ x < 2π. 522 Chapter 9 hsnb_alg2_pe_0908.indd 522 Trigonometric Ratios and Functions 2/5/15 1:55 PM 9.8 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Write the expression cos 130° cos 40° − sin 130° sin 40° as the cosine of an angle. 2. WRITING Explain how to evaluate tan 75° using either the sum or difference formula for tangent. Monitoring Progress and Modeling with Mathematics In Exercises 3–10, find the exact value of the expression. (See Example 1.) 3. tan(−15°) 24. 4. tan 195° 23π 12 5. sin — 17π 12 11. sin(a + b) 12. sin(a − b) 13. cos(a − b) 14. cos(a + b) 15. tan(a + b) 16. tan(a − b) for 0 ≤ x < 2π? π A — ○ 3 2π C — ○ 3 π 18. cos x − — 2 19. cos(x + 2π) 3π 2 21. sin x − — ( π 2 22. tan x + — ) π tan x + tan — π 4 tan x + — = —— π 4 1 + tan x tan — 4 tan x + 1 =— 1 + tan x ) 29. 30. π 28. tan( x − ) = 0 ( π2 ) 12 4 π π cos( x + ) − cos( x − ) = 1 6 6 π π sin( x + ) + sin( x − ) = 0 4 4 — — — — — 31. tan(x + π) − tan(π − x) = 0 32. sin(x + π) + cos(x + π) = 0 33. USING EQUATIONS Derive the cofunction identity π sin — − θ = cos θ using the difference formula 2 for sine. ( =1 Section 9.8 hsnb_alg2_pe_0908.indd 523 7π D — ○ 4 27. sin x + — = — ) correct the error in simplifying the expression. 3π B — ○ 4 In Exercises 27– 32, solve the equation for 0 ≤ x < 2π. (See Example 4.) ERROR ANALYSIS In Exercises 23 and 24, describe and ( 5π D — ○ 6 for 0 ≤ x < 2π ? π A — ○ 4 5π C — ○ 4 20. tan(x − 2π) ) π B — ○ 6 26. What are the solutions of the equation tan x + 1 = 0 In Exercises 17–22, simplify the expression. (See Example 3.) ( — 25. What are the solutions of the equation 2 sin x − 1 = 0 In Exercises 11–16, evaluate the expression given π 15 4 that cos a = — with 0 < a < — and sin b = −— with 5 2 17 3π < b < 2π. (See Example 2.) — 2 ✗ — √2 √2 = — cos x − — sin x 2 2 — ( ) 17. tan(x + π) ) √2 = — (cos x − sin x) 2 11π 8. cos — 12 7π 10. sin −— 12 9. tan — 23. ( π π π sin x − — = sin — cos x − cos — sin x 4 4 4 6. sin(−165°) 7. cos 105° ( ✗ ) Using Sum and Difference Formulas 523 2/5/15 1:55 PM 34. MAKING AN ARGUMENT Your friend claims it is 38. HOW DO YOU SEE IT? Explain how to use the figure π π to solve the equation sin x + — − sin — − x = 0 4 4 for 0 ≤ x < 2π. ( possible to use the difference formula for tangent to π derive the cofunction identity tan — − θ = cot θ. Is 2 your friend correct? Explain your reasoning. ( ) y ( f(x) = sin x + 35. MODELING WITH MATHEMATICS A photographer is at a height h taking aerial photographs with a 35-millimeter camera. The ratio of the image length WQ to the length NA of the actual object is given by the formula WQ NA 35 tan(θ − t) + 35 tan t h tan θ h — = —— ( π 4 ) ( x π −1 camera θ ) g(x) = sin Q 2π ( π4 − x( t W 39. MATHEMATICAL CONNECTIONS The figure shows the N acute angle of intersection, θ2 − θ1, of two lines with slopes m1 and m2. A where θ is the angle between the vertical line perpendicular to the ground and the line from the camera to point A and t is the tilt angle of the film. When t = 45°, show that the formula can be rewritten 70 WQ as — = ——. (See Example 5.) NA h(1 + tan θ) y y = m1x + b1 y = m2 x + b2 θ 2 − θ1 36. MODELING WITH MATHEMATICS When a wave travels through a taut string, the displacement y of each point on the string depends on the time t and the point’s position x. The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula 2πt 2πx 2πt 2πx y = A cos — − — + A cos — + — . 3 5 3 5 When t = 1, show that the formula can be rewritten as 2πx y = −A cos — . 5 ( ) ( θ1 x a. Use the difference formula for tangent to write an equation for tan (θ2 − θ1) in terms of m1 and m2. ) 37. MODELING WITH MATHEMATICS The busy signal on a touch-tone phone is a combination of two tones with frequencies of 480 hertz and 620 hertz. The individual tones can be modeled by the equations: θ2 b. Use the equation from part (a) to find the acute angle of intersection of the lines y = x − 1 and — 4 − √3 1 y= — x+— — —. √3 − 2 2 − √3 ( ) 40. THOUGHT PROVOKING Rewrite each function. Justify your answers. a. Write sin 3x as a function of sin x. 480 hertz: y1 = cos 960πt b. Write cos 3x as a function of cos x. 620 hertz: y2 = cos 1240πt c. Write tan 3x as a function of tan x. The sound of the busy signal can be modeled by y1 + y2. Show that y1 + y2 = 2 cos 1100πt cos 140πt. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Solve the equation. Check your solution(s). (Section 7.5) 9 x−2 7 2 41. 1 − — = −— 524 Chapter 9 hsnb_alg2_pe_0908.indd 524 12 x 3 4 8 x 42. — + — = — 2x − 3 x+1 10 x −1 +5 43. — = — 2 Trigonometric Ratios and Functions 2/5/15 1:55 PM 9.5–9.8 What Did You Learn? Core Vocabulary frequency, p. 506 sinusoid, p. 507 trigonometric identity, p. 514 Core Concepts Section 9.5 Characteristics of y = tan x and y = cot x, p. 498 Period and Vertical Asymptotes of y = a tan bx and y = a cot bx, p. 499 Characteristics of y = sec x and y = csc x, p. 500 Section 9.6 Frequency, p. 506 Writing Trigonometric Functions, p. 507 Using Technology to Find Trigonometric Models, p. 509 Section 9.7 Fundamental Trigonometric Identities, p. 514 Section 9.8 Sum and Difference Formulas, p. 520 Trigonometric Equations and Real-Life Formulas, p. 522 Mathematical Practices 1. Explain why the relationship between θ and d makes sense in the context of the situation in Exercise 43 on page 503. 2. How can you use definitions to relate the slope of a line with the tangent off an ang angle glee iin n Exercise 39 on page 524? Performance Task Lightening the Load You need to move a heavy table across the room. What is the easiestt way to move it? Should you push it? Should you tie a rope around one nee leg of the table and pull it? How can trigonometry help you make the e right decision? To explore the answers to these questions and more, go to BigIdeasMath.com. 525 9 Chapter Review 9.1 Dynamic Solutions available at BigIdeasMath.com Right Triangle Trigonometry (pp. 461−468) Evaluate the six trigonometric functions of the angle θ. From the Pythagorean Theorem, the length of the hypotenuse is 6 — hyp. = √62 + 82 — θ = √100 8 = 10. Using adj. = 8, opp. = 6, and hyp. = 10, the values of the six trigonometric functions of θ are: 6 3 opp. sin θ = — = — = — hyp. 10 5 8 4 adj. cos θ = — = — = — hyp. 10 5 opp. 6 3 tan θ = — = — = — adj. 8 4 hyp. 10 5 csc θ = — = — = — opp. 6 3 hyp. 10 5 sec θ = — = — = — adj. 8 4 8 4 adj. cot θ = — = — = — opp. 6 3 6 1. In a right triangle, θ is an acute angle and cos θ = — . Evaluate the other five trigonometric 11 functions of θ. 2. The shadow of a tree measures 25 feet from its base. The angle of elevation to the Sun is 31°. How tall is the tree? 31° 25 ft 9.2 Angles and Radian Measure (pp. 469−476) Convert the degree measure to radians or the radian measure to degrees. 7π b. — 12 a. 110° π radians 110° = 110 degrees — 180 degrees ( ) 11π =— 18 7π 12 7π 12 180° ( π radians ) — = — radians — = 105° 3. Find one positive angle and one negative angle that are coterminal with 382°. Convert the degree measure to radians or the radian measure to degrees. 4. 30° 5. 225° 3π 4 6. — 5π 3 7. — 8. A sprinkler system on a farm rotates 140° and sprays water up to 35 meters. Draw a diagram that shows the region that can be irrigated with the sprinkler. Then find the area of the region. 526 Chapter 9 hsnb_alg2_pe_09ec.indd 526 Trigonometric Ratios and Functions 2/5/15 1:44 PM 9.3 Trigonometric Functions of Any Angle (pp. 477−484) Evaluate csc 210°. The reference angle is θ′ = 210° − 180° = 30°. The cosecant function is negative in Quadrant III, so csc 210° = −csc 30° = −2. Evaluate the six trigonometric functions of θ. 9. y 10. (0, 1) 11. y y (−4, 6) θ θ θ x x x (24, −7) Evaluate the function without using a calculator. 12. tan 330° 9.4 13π 6 13. sec(−405°) 11π 3 14. sin — 15. sec — Graphing Sine and Cosine Functions (pp. 485−494) 1 Identify the amplitude and period of g(x) = — sin 2x. Then graph the function and describe the 2 graph of g as a transformation of the graph of f (x) = sin x. 1 The function is of the form g(x) = a sin bx, where a = — and b = 2. So, the amplitude is 2 1 2π 2π a = — and the period is — = — = π. y 2 b 2 π 1 Intercepts: (0, 0); — π, 0 = —, 0 ; (π, 0) 2 2 ( ) ( ) π 1 1 1 Maximum: ( ⋅ π, ) = ( , ) 4 2 4 2 3 1 3π 1 Minimum: ( ⋅ π, − ) = ( , − ) 4 2 4 2 — — ⋅ — — — — — 0.5 π 4 3π 4 x −0.5 — The graph of g is a vertical shrink by a factor of —12 and a horizontal shrink by a factor of —12 of the graph of f. Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of the parent function. 1 16. g(x) = 8 cos x 17. g(x) = 6 sin πx 18. g(x) = — cos 4x 4 Graph the function. 19. g(x) = cos(x + π) + 2 20. g(x) = −sin x − 4 ( Chapter 9 hsnb_alg2_pe_09ec.indd 527 π 2 21. g(x) = 2 sin x + — ) Chapter Review 527 2/5/15 1:44 PM 9.5 Graphing Other Trigonometric Functions (pp. 497−504) a. Graph one period of g(x) = 7 cot πx. Describe the graph of g as a transformation of the graph of f (x) = cot x. π π The function is of the form g(x) = a cot bx, where a = 7 and b = π. So, the period is — = — = 1. ∣b∣ π π π 1 Intercepts: —, 0 = —, 0 = —, 0 2b 2π 2 ( ) ( ) ) ( y 7 π π Asymptotes: x = 0; x = — = —, or x = 1 ∣b∣ π π π 1 Halfway points: —, a = —, 7 = —, 7 ; 4b 4π 4 ( ) ( ) ( ) 1 2 −7 1 x ( 34bπ, −a ) = ( 43ππ, −7 ) = ( 34, −7 ) — — — The graph of g is a vertical stretch by a factor of 7 and a horizontal shrink by a 1 factor of — of the graph of f. π b. Graph one period of g(x) = 9 sec x. Describe the graph of g as a transformation of the graph of f (x) = sec x. Step 1 Graph the function y = 9 cos x. 2π The period is — = 2π. 1 y 18 Step 2 Graph asymptotes of g. Because the asymptotes of g occur when 9 cos x = 0, π π 3π graph x = −—, x = —, and x = —. 2 2 2 π 2 π − 2 x −18 Step 3 Plot the points on g, such as (0, 9) and (π, −9). Then use the asymptotes to sketch the curve. The graph of g is a vertical stretch by a factor of 9 of the graph of f. Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. 1 2 22. g(x) = tan —x 24. g(x) = 4 tan 3πx 23. g(x) = 2 cot x Graph the function. 528 26. g(x) = sec —x 27. g(x) = 5 sec πx 28. g(x) = — csc —x Chapter 9 hsnb_alg2_pe_09ec.indd 528 1 2 25. g(x) = 5 csc x 1 2 π 4 Trigonometric Ratios and Functions 2/5/15 1:44 PM 9.6 Modeling with Trigonometric Functions (pp. 505−512) Write a function for the sinusoid shown. Step 1 Step 2 (π2 , 3( y 4 Find the maximum and minimum values. From the graph, the maximum value is 3 and the minimum value is −1. Identify the vertical shift, k. The value of k is the mean of the maximum and minimum values. π 2 7π 6 (π6 , −1( −2 x (maximum value) + (minimum value) 3 + (−1) 2 k = ———— = — = — = 1 2 2 2 Step 3 Decide whether the graph should be modeled by a sine or cosine function. Because the graph crosses the midline y = 1 on the y-axis and then decreases to its minimum value, the graph is a sine curve with a reflection in the x-axis and no horizontal shift. So, h = 0. Step 4 Find the amplitude and period. 2π 2π The period is — = —. So, b = 3. 3 b The amplitude is (maximum value) − (minimum value) 3 − (−1) 4 ∣ a ∣ = ———— = — = — = 2. 2 2 2 Because the graph is a reflection in the x-axis, a < 0. So, a = −2. The function is y = −2 sin 3x + 1. Write a function for the sinusoid. 29. y −π 30. (3π , 1) 1 π y −3 (0, −1) 1 3 x x 3π (π , −1) −4 (1, −3) 31. You put a reflector on a spoke of your bicycle wheel. The highest point of the reflector is 25 inches above the ground, and the lowest point is 2 inches. The reflector makes 1 revolution per second. Write a model for the height h (in inches) of a reflector as a function of time t (in seconds) given that the reflector is at its lowest point when t = 0. 32. The table shows the monthly precipitation P (in inches) for Bismarck, North Dakota, where t = 1 represents January. Write a model that gives P as a function of t and interpret the period of its graph. t 1 2 3 4 5 6 7 8 9 10 11 12 P 0.5 0.5 0.9 1.5 2.2 2.6 2.6 2.2 1.6 1.3 0.7 0.4 Chapter 9 hsnb_alg2_pe_09ec.indd 529 Chapter Review 529 2/5/15 1:44 PM 9.7 Using Trigonometric Identities (pp. 513−518) cot2 θ Verify the identity — = csc θ − sin θ. csc θ cot2 θ csc θ csc2 θ − 1 csc θ —=— Pythagorean identity csc2 θ 1 =—−— csc θ csc θ Write as separate fractions. 1 = csc θ − — csc θ Simplify. = csc θ − sin θ Reciprocal identity Simplify the expression. (sec x + 1)(sec x − 1) tan x 33. cot2 x − cot2 x cos2 x 34. —— ( π2 ) 35. sin — − x tan x Verify the identity. ( π2 ) cos x sec x 1 + tan x 37. tan — − x cot x = csc2 x − 1 36. — = cos2 x 2 9.8 Using Sum and Difference Formulas (pp. 519−524) Find the exact value of sin 105°. sin 105° = sin(45° + 60°) Substitute 45° + 60° for 105°. = sin 45° cos 60° + cos 45° sin 60° — — √2 1 √2 √3 =— —+— — 2 2 2 2 — ⋅ ⋅ Sum formula for sine — Evaluate. — √2 + √6 =— 4 Simplify. — — √2 + √6 The exact value of sin 105° is —. 4 Find the exact value of the expression. 38. sin 75° π 12 39. tan(−15°) 1 4 40. cos — 3π 2 π 2 3 7 41. Find tan(a + b), given that tan a = — with π < a < — and tan b = — with 0 < b < —. Solve the equation for 0 ≤ x < 2π. ( 3π 4 ) ( 3π 4 ) 42. cos x + — + cos x − — = 1 530 Chapter 9 hsnb_alg2_pe_09ec.indd 530 ( π 2 ) 43. tan(x + π) + cos x + — = 0 Trigonometric Ratios and Functions 2/5/15 1:44 PM 9 Chapter Test Verify the identity. cos2 x + sin2 x 1 + tan x 1 + sin x cos x 3π 2 ( cos x 1 + sin x 2. — + — = 2 sec x 1. —— = cos2 x 2 ) 3. cos x + — = sin x 4. Evaluate sec(−300°) without using a calculator. Write a function for the sinusoid. 5. y 6. (2, 5) ( 98π , 1( y 1 π 4 3 π 2 π 5π 4 x −3 1 2 (1, −1) (38π , −5( −5 x Graph the function. Then describe the graph of g as a transformation of the graph of its parent function. 1 7. g(x) = −4 tan 2x 8. g(x) = −2 cos —x + 3 9. g(x) = 3 csc πx 3 Convert the degree measure to radians or the radian measure to degrees. Then find one positive angle and one negative angle that are coterminal with the given angle. 4π 5 10. −50° 8π 3 11. — 12. — 13. Find the arc length and area of a sector with radius r = 13 inches and central angle θ = 40°. Evaluate the six trigonometric functions of the angle θ. 14. 15. y y θ θ x x (−1, 0) (2, −9) 16. In which quadrant does the terminal side of θ lie when cos θ < 0 and tan θ > 0? Explain. 200 ft 17. How tall is the building? Justify your answer. h 60° 18. The table shows the average daily high temperatures T (in degrees Fahrenheit) in 5 ft Baltimore, Maryland, where m = 1 represents January. Write a model that gives T as a function of m and interpret the period of its graph. m 1 2 3 4 5 6 7 8 9 10 11 12 T 41 45 54 65 74 83 87 85 78 67 56 45 Chapter 9 hsnb_alg2_pe_09ec.indd 531 Not drawn to scale Chapter Test 531 2/5/15 1:44 PM 9 Cumulative Assessment 1. Which expressions are equivalent to 1? tan x sec x cos x π cos — − x csc x 2 cos2(−x) tan2 x sin (−x) sin2 x + cos2 x ( —— 2 ) 2. Which rational expression represents the ratio of the perimeter to the area of the playground shown in the diagram? 9 A — ○ 7x 11 B — ○ 14x 2x yd x yd 2x yd 1 C — ○ x 6x yd 1 D — ○ 2x 3. The chart shows the average monthly temperatures (in degrees Fahrenheit) and the gas usages (in cubic feet) of a household for 12 months. a. Use a graphing calculator to find trigonometric models for the average temperature y1 as a function of time and the gas usage y2 (in thousands of cubic feet) as a function of time. Let t = 1 represent January. January February March April 32°F 21°F 15°F 22°F 20,000 ft3 27,000 ft3 23,000 ft3 22,000 ft3 May June July August 35°F 49°F 62°F 78°F 21,000 ft3 14,000 ft3 8,000 ft3 9,000 ft3 September October November December 71°F 63°F 55°F 40°F 13,000 ft3 15,000 ft3 19,000 ft3 23,000 ft3 b. Graph the two regression equations in the same coordinate plane on your graphing calculator. Describe the relationship between the graphs. 4. Evaluate each logarithm using log2 5 ≈ 2.322 and log2 3 ≈ 1.585, if necessary. Then order the logarithms by value from least to greatest. a. log 1000 b. log2 15 c. ln e d. log2 9 e. 532 log2 —53 Chapter 9 hsnb_alg2_pe_09ec.indd 532 f. log2 1 Trigonometric Ratios and Functions 2/5/15 1:44 PM 5. Which function is not represented by the graph? A y = 5 sin x ○ π B y = 5 cos — − x ○ 2 ( π C y = 5 cos x + — ○ 2 ( y ) g 5 ) x 3π − 2 π 2 D y = −5 sin(x + π) ○ 6. Complete each statement with < or > so that each statement is true. a. θ b. tan θ c. θ′ y 3 radians s = 4π 0 θ r=6 x 45° 7. Use the Rational Root Theorem and the graph to find all the real zeros of the function f (x) = 2x3 − x2 − 13x − 6. y f 5 2 x −10 −20 5π 6 8. Your friend claims −210° is coterminal with the angle —. Is your friend correct? Explain your reasoning. 9. Company A and Company B offer the same starting annual salary of $20,000. Company A gives a $1000 raise each year. Company B gives a 4% raise each year. a. Write rules giving the salaries an and bn for your nth year of employment at Company A and Company B, respectively. Tell whether the sequence represented by each rule is arithmetic, geometric, or neither. b. Graph each sequence in the same coordinate plane. c. Under what conditions would you choose to work for Company B? d. After 20 years of employment, compare your total earnings. Chapter 9 hsnb_alg2_pe_09ec.indd 533 Cumulative Assessment 533 2/5/15 1:44 PM

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