SHARP TRIG COVER 02.2.19 10:55 AM Page 3 APPLYING TRIGONOMETRY SHARP U S I N G T H E EGraphing L - 9Calculator 600 D A V I D P. L A W R E N C E Applying TRIGONOMETRY using the SHARP EL-9600 GRAPHING CALCULATOR David P. Lawrence Southwestern Oklahoma State University This Teaching Resource has been developed specifically for use with the Sharp EL-9600 graphing calculator. The goal for preparing this book was to provide mathematics educators with quality teaching materials that utilize the unique features of the Sharp graphing calculator. This book, along with the Sharp graphing calculator, offers you and your students 10 classroom-tested, topic-specific lessons that build skills. Each lesson includes Introducing the Topic, Calculator Operations, Method of Teaching, explanations for Using Blackline Masters, For Discussion, and Additional Problems to solve. Conveniently located in the back of the book are 36 reproducible Blackline Masters. You’ll find them ideal for creating handouts, overhead transparencies, or to use as student activity worksheets for extra practice. Solutions to the Activities are also included. We hope you enjoy using this resource book and the Sharp El-9600 graphing calculator in your classroom. Other books are also available: Applying STATISTICS using the SHARP EL-9600 Graphing Calculator Applying PRE-ALGEBRA and ALGEBRA using the SHARP EL-9600 Graphing Calculator Applying PRE-CALCULUS and CALCULUS using the SHARP EL-9600 Graphing Calculator Graphing Calculators: Quick & Easy! The SHARP EL-9600 TRIGONOMETRY USING THE SHARP EL-9600 i Dedicated to my grandma, Marjorie Lee Special thanks to Ms. Marina Ramirez and Ms. Melanie Drozdowski for their comments and suggestions. Developed and prepared by Pencil Point Studio. Copyright © 1998 by Sharp Electronics Corporation. All rights reserved. This publication may not be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without written permission. The blackline masters in this publication are designed to be used with appropriate duplicating equipment to reproduce for classroom use. First printed in the United States of America in 1998. ii TRIGONOMETRY USING THE SHARP EL-9600 CONTENTS CHAPTER TOPIC PAGE 1 Graphs of Trigonometric Functions 1 2 Evaluating Trigonometric Functions 5 3 Graphical Verification of Formulas and Identities 10 4 Amplitude, Period and Phase Shift 14 5 Inverse Trigonometric Functions 19 6 Solving Trigonometric Equations 23 7 Applications with Right Triangles 29 8 Law of Sines and Law of Cosines 34 9 Area of a Triangle 41 Area of a Circular Sector 47 Blackline Masters 52 Solutions to the Activities 89 10 TRIGONOMETRY USING THE SHARP EL-9600 iii Chapter one GRAPHS OF TRIGONOMETRIC FUNCTIONS Introducing the Topic In this chapter, you and your students will learn how to graph trigonometric functions on the Sharp EL-9600. Graphing trigonometric functions on the calculator is covered first, so that graphical evaluation of trigonometric functions may be covered in the next chapter. The trigonometric functions are often introduced as circular functions with angle measurement in degrees and radians. The sine and cosine functions are said to map the angle measurement to the y and x coordinates respectively, of the point on the unit circle representing the angle. Then, the four additional trigonometric functions are defined in terms of sine and cosine. If you let the angle measurement be represented by a horizontal axis (x) and let the vertical axis (y) represent the function evaluated at the angle, then you can graph the six trigonometric functions as y = sin x, y = cos x, y = tan x, y = csc x, y = sec x, and y = cot x. Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 1 Calculator Operations Turn the calculator on and press Y= . The Y prompts will appear in the viewing window. Press CL to remove an old Y1 expression. Press ▼ CL to clear additional expressions. The calculator should be setup in radian measurement, with rectangular coordinates, and in equation editor mode. To complete this setup, press 2ndF SET UP , touch B DRG, double touch 2 Rad, touch E COORD, double touch 1 Rect, touch G EDITOR, and double touch 1 Equation. Press 2ndF QUIT again to exit the menu and return to the Y prompts. To enter the sine function ( y = sin x ) for Y1, press sin X/θ/T/n . Enter the viewing window range by pressing ZOOM , touching E TRIG, and double touching 1 sin x. This establishes built-in viewing-window range for graphing the sine function. The window ranges are -2π ≤ x ≤ 2π and -1.55 ≤ y ≤ 1.55. The scale for the x axis is π and the y-axis scale is .5. 2 The remaining five trigonometric functions can be graphed similarly. 2 Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 Method of Teaching Use the Blackline Master 1.1 to create an overhead for graphing the sine function. Go over in detail how to graph a trigonometric function. Follow the sine function demonstration with the cosecant function demonstration provided in the sample problem below and on the Blackline Master. Next, use the Blackline Masters 1.2 through 1.4 to create worksheets for the students. Have the students graph the remaining four trigonometric functions. ▼ feature by pressing TRACE . Move the tracer by pressing or ▼ Use the topics For Discussion to supplement the worksheets. Engage the trace . Tracing will assist you in discussing the topics. Using Blackline Master 1.1 The problem discussed above in the Calculator Operations section is presented on Blackline Master 1.1. The following problem also appears on the Blackline Master. Graph the cosecant function by pressing Y= CL to remove an old Y1 expression. To enter the cosecant function for Y1, press MATH , you may need to touch A CALC if the menu is not already there. Double touch 10 csc, and press X/θ/T/n . Use the sine function's built-in viewing range, and press GRAPH to view the graph. Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 3 For Discussion You and your students can use the calculator to discuss: (1) intervals where the functions increase and decrease, (2) domain and range of the functions, (3) the periodicity of the functions, and (4) which part of the horizontal axis represents each quadrant. The trace feature will allow you to move along the curve to see where the function is increasing and decreasing, to see the values associated with limits on the domain and range, to see the period of a function, and to explore the quadrant "markers" on the x-axis. 4 Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 Chapter two EVALUATING TRIGONOMETRIC FUNCTIONS Introducing the Topic In this chapter, you and your students will learn how to evaluate trigonometric functions directly in radians or degrees, and graphically through the tracing feature. Calculator Operations To evaluate sin 45° directly, you will turn the calculator on and press + – × ÷ to enter computation mode. Press CL to clear the viewing window of any old expressions. To evaluate trigonometric functions in degrees, you can set the calculator to operate in degrees by pressing 2ndF and double touch 1 Deg. Press 2ndF sin 45° you will press sin 4 5 SET UP , touch B DRG, QUIT to exit the menu. Now, to find the ENTER . Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 5 To approximate sin 45° graphically, you will press Y= followed by CL to clear an old expression for Y1. Enter the function sin x by pressing sin X/θ/T/n . To view the graph press ZOOM , touch E TRIG, and double touch 1 sin x. x = 45° by pressing ▼ Now, engage the trace by pressing TRACE and moving the tracer near . With the cursor located near x = 45°, you can zoom in to find a better approximate for the y value. First, set the factors for zooming by pressing ZOOM , touching B FACTOR, and pressing ENTER 5 ENTER 5 ENTER . These larger values will accelerate the zooming in. Now, press ZOOM , touch A ZOOM, and double touch 3 In to zoom in on the cursor. Once again, trace over near x = 45°. Zoom in two more times to see a better approximation. π + To find sin 3 directly, you will turn the calculator on and press × –÷ to enter computation mode. Press CL to clear the viewing window of any old expressions. To evaluate trigonometric functions in radian mode, you can set the calculator to operate in radians by pressing 2ndF SET UP , touching B DRG, and double touching 2 Rad. Press 2ndF QUIT to exit the menu. Now, to find sin π you will press sin 2ndF π a/b 3 ENTER . 3 6 Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 π graphically, you will press Y= followed by CL to clear 3 an old expression for Y1. Enter the function sin x by pressing sin X/θ/T/n . To approximate sin View the graph by pressing ZOOM , touching E TRIG, double touching 1 sin x. π Now, engage the trace by pressing TRACE and tracing near x = = 1.047. 3 With the cursor located near x = 1.047, you can zoom in to find a better approximation for the y value. Press ZOOM , touch A ZOOM, and double touch 3 In to zoom in on the cursor. Once again, trace over near x = 1.047. Zoom in one more time to see an even better approximation. Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 7 Method of Teaching Use Blackline Master 2.1 to create an overhead or handout for direct and graphical evaluation of trigonometric functions. Demonstrate the procedure changes for radian measurement. Remind the students of the location of cosecant, secant, and cotangent within the MATH menu. Next, use the Blackline Masters 2.2 through 2.4 to create worksheets for the students. Have the students evaluate the six trigonometric functions directly and graphically for the various degree and radian measurements. Use the topics For Discussion to supplement the worksheets. Using Blackline Master 2.2 π The following problem appears on Blackline Master 2.2. Evaluate sec 4 directly in degree mode. Press +× –÷ CL to enter computation mode and clear the home screen. Press 2ndF SET UP to verify the calculator is in degree mode. If it is not, change it to degree mode. Press 2ndF QUIT to exit the menu. Enter and 09 sec, pressing 2ndF π a/b 4 ▼ evaluate the expression by pressing MATH , touching A CALC, double touching MATH , touching E ANGLE, double touching 4 r, and pressing ENTER . Evaluate tan 20° graphically. Press Y= and press CL to remove an old expression for Y1. Press 2ndF SET UP to verify the calculator is in degree mode. If it is not, change it. Press 2ndF function tan x for Y1 by pressing tan QUIT to exit the menu. Enter the X/θ/T/n . Instead of graphing the function in the default viewing window for tangent, press WINDOW adjust the Xmin and Xmax to 10° and 30° (value of interest ± 10° ) respectively to start out closer to the value of interest. Set the Xscl = 1. Press ZOOM , touch A ZOOM, 8 Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 and double touch 1 Auto to view the graph. Now, engage the trace by pressing TRACE . Notice the tracer appears at x = 20° since it is the middle of the x values. For Discussion Students can calculate sin 25° in radian mode by pressing sin 2 5 MATH , touching E ANGLE, double touching 1°, and pressing ENTER . Students can store a graph with a particular viewing window for repeated use by using the StoGD command in the DRAW menu. For more detail, refer to the EL-9600 Owner's Manual. Discuss the use of the negative sign like cos (–10°). This is the first assignment that uses a negative angle. Also remind them of the difference between the negative key (a short dash within parentheses) and the minus key (no parentheses). Additional Problems Evaluate the following directly: (1) cos 27° (2) cot 52° 5π (3) sec 3 Evaluate the following graphically: (4) sin 107° π (5) tan 7 5π (6) csc 4 Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 9 Chapter three GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES Introducing the Topic In this chapter, you and your students will learn how to verify trigonometric formulas and identities graphically. The left and right sides of an equality can be graphed by the calculator as Y1 and Y2. Verification is made by viewing Y1 and Y2 as the same graph, hence equivalent expressions. Calculator Operations To verify the double angle formula sin 2θ = 2 sin θ cos θ, you will graph the "sin 2θ" as Y1 = sin (2x) and the "2 sin θ cos θ" as Y2 = 2 sin x cos x and show that they are in fact the same graph. To do this you will turn the calculator on and press Y= . Press CL to clear Y1 of an old expression. Press ▼ CL to clear additional expressions. Make sure the calculator is in radian mode by pressing 2ndF SET UP , touch B DRG, and double touch 2 Rad. Press 2ndF QUIT to exit the menu. Now enter sin (2x) for Y1 by pressing sin ( and the 2 sin x cos x for Y2 by pressing 2 sin 2 X/θ/T/n X/θ/T/n on the screen, touch 6 2π<X,<2π, touch 10 ➧ – cos ENTER X/θ/T/n EZ , touch 5, touch ➧ ENTER . Set your viewing window by pressing WINDOW ) on the screen, and touch 5 –5<Y<5. Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 To verify the identity sin2 θ + cos2 θ = 1, you will graph the "sin2 θ + cos2 θ" as Y1 = sin2 x + cos2 x and the "1" as Y2 = 1, and show that they are the same graph. To do this press Y= to access the Y1 prompt. Press CL to clear Y1 of an old expression, and enter the sin2 x + cos2 x for Y1 by pressing ( b sin ▼ b + ( 1 for Y2 by pressing CL 1 ENTER . Press GRAPH to view the graph. X/θ/T/n ) a 2 cos X/θ/T/n ) a 2 ENTER . Enter the Method of Teaching Use Blackline Master 3.1 to create an overhead or handout for graphically verifying trigonometric formulas and identities. Demonstrate the process of entering the equality's left- and right-hand expressions and when necessary, emphasize the intervals where functions are positive or negative (as demonstrated in Using Blackline Master 3.2). Next, use the Blackline Masters 3.3 through 3.5 to create worksheets for the students. Have the students verify the various formulas and identities. Use the topics For Discussion to supplement the worksheets. Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 11 Using Blackline Master 3.2 The following problem appears on Blackline Master 3.2. Verify the half angle θ formula cos 2 = ±√(1 + cos θ2 ). To verify the formula, you will graph the “cos θ x √ 1 + cos θ ” as Y2 = ± √ 1 + cos θ and show that they 2 ” as Y1 = cos 2 and the “± 2 2 are in fact the same graph. In this problem, you will need to limit your domain or x values to intervals where cosine is either negative or positive and enter the appropriate radical for Y2. To do this, you will turn the calculator on and press Y= . Press CL to clear Y1 of an old expression. x Now enter the cos 2 for Y1 by pressing cos X/θ/T/n a/b 2 ENTER and 1 + cos x the for Y2 (limiting to where cosine is positive) by pressing CL 2 x 2ndF √ a/b 1 + cos X/θ/T/n ▼ 2 ENTER . Cos 2 will be positive for a viewing window of 0 < x < π . Enter this viewing window by pressing WINDOW 0 ENTER 2ndF π ENTER 2ndF π ÷ 2 ENTER (–) 3 ENTER 3 ENTER 1 ENTER . Press GRAPH to view the graphs. Notice they appear as one graph. To verify the formula for where cosine is negative, press Y= ▼ to access the x Y2 prompt and press (–) to insert a negative in front of the radical. Cos 2 will be negative for a viewing window of π< x < 3π. Enter this viewing window by pressing WINDOW 2ndF π ENTER 3 2ndF π ENTER . Press GRAPH to view the graphs. Notice they appear as one graph. 12 Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 For Discussion Students can use the calculators to discuss how "toggling" (using the ▲ and ▼ keys while tracing) between Y1 and Y2 verifies the same values for both Y1 and Y2, and they can explore what happens when they do not limit the domain on those problems with ± parts (they see two graphs). Additional Problems Verify the following formulas and identities graphically. 1. cos 2θ = cos2 θ – sin2 θ 2. 1 + cot2 θ = csc2 θ 3. sin2 θ cot2 θ = (1 + sin θ)(1 – sin θ) Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 13 Chapter four AMPLITUDE, PERIOD, AND PHASE SHIFT Introduction In this chapter, you and your students will learn how to modify trigonometric functions by constants. If you multiply sine or cosine by a constant A, then every value of the function is multiplied by A. The maximum absolute value of the sine or cosine function is 1. Therefore, Asin θ and Acos θ will have a maximum value of |A|. The value of |A| is called the amplitude of the functions y = Asin x and y = Acos x, and is the distance sine and cosine rises and falls above the x-axis. Multiplication of the function by A will stretch or shrink the curve vertically. |A| > 1 will stretch the curve while an |A| between 0 and 1 will shrink the curve. The tangent function does not have an amplitude because π tangent increases without bound as θ approaches values such as 2 . If we multiply the argument θ by a constant k within a trigonometric function, then every measure of an angle θ is multiplied by k. Multiplication of θ will stretch or shrink the curve horizontally. A value of k > 1 will shrink the curve while a value of k between 0 and 1 will stretch the curve. The stretching and shrinking of a curve horizontally effects the function's period. The new periods π 2π will be for sine, cosine, secant and cosecant, and k for tangent and cotangent. k 14 Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 If we add a constant c to the argument θ within a trigonometric function, then every measure of an angle θ is increased by c. Addition of c will shift the function left or right. A c > 0 will shift the function to the left while a c < 0 will shift the function to the right. If we combine both the multiplication by a constant k and the addition of a constant c to the argument θ (sin (kθ + c)), then the movement of the function left or right with respect to the period change is called the phase shift. The -c c value for this shift is . A c > 0 will shift the function units to the left while a k k c c < 0 will shift the function units to the right. k Calculator Operations To investigate changes in amplitude, you will use the SHIFT/CHANGE feature of the graphing calculator. Press 2ndF SHIFT/CHANGE , touch B CHANGE, and double touch 5 Y=sin X. Double touch the screen at Y = 3, above the box with the arrows. Notice the new function Y = 3 sin X is displayed. Press ENTER to place the new function in the list. Double touch the screen at Y = -2, below the box with the arrows. Notice the new function Y = -2 sin X is displayed. Press ENTER to place the new function in the list. Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 15 Touch the top of the right side of the screen and the list of three functions will be displayed. The function Y = sin X is highlighted on the right side of the screen, and the graph is darkened on the left side of the screen. Press ▼ to move from one graph to the next. Observe and note the changes in sin X when you change the amplitude. Method of Teaching Use Blackline Master 4.1 to create an overhead or handout for demonstrating changes in amplitude. Next use Blackline Masters 4.2 and 4.3 to create worksheets to demonstrate changes in period and phase shift. For analysis of period, demonstrate the process of entering a "standard" trigonometric function for Y1 and a changed function for Y2. Use the trace feature to verify the new period. For analysis of phase shift, demonstrate the use of SHIFT feature. Use the topics For Discussion to supplement the worksheets. Using Blackline Masters 4.2 and 4.3 The following problems appear on Blackline Masters 4.2 and 4.3. To investigate the change in a trigonometric function by multiplying the argument θ by 2, press Y= CL to access and clear Y1 of an old expression. Make sure the calculator is in radian mode by pressing 2ndF 2 Rad. Press 2ndF SET UP , touch B DRG, and double touch QUIT to exit the menu. Now, enter sin x for Y1 by pressing sin Y2 by pressing CL sin ( 2 X/θ/T/n X/θ/T/n ) ENTER and the sin (2x) for ENTER . Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 1 sin x. Note the change in the second graph versus the first graph. Engage the trace to show the period has changed from 2π to π. 16 Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 3π to the 2 SHIFT/CHANGE , touch A SHIFT, and double touch To investigate the change in a trigonometric function by subtracting argument θ, press 2ndF 6 Y=sin X. Double touch the screen at X = 3π/2 to the right of the box with the arrows. Notice the new function Y = sin (X - 3π/2) is displayed. Press ENTER to place the new function in the list. Double touch the screen at Y = -2, below the box with the arrows. Notice the new function Y = sin (X - 3π/2) – 2 is displayed. Press ENTER to place the new function in the list. Touch the top of the right side of the screen and the list of three functions will be displayed. The function Y = sin X is highlighted on the right side of the screen, and the graph is darkened on the left side of the screen. Press ▼ to move from one graph to the next. Observe and note the changes in sin X when you change the phase shift. Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 17 For Discussion Students can use the calculators to discuss why tangent does not have an amplitude, the period and period changes of tangent, the phase shifts for tangent, and what happens when you multiply tangent by a constant. Additional Problems Use the following to demonstrate the period, period changes and phase shifts for tangent. (1) Compare tan θ with (-2)tan θ (2) Compare tan θ with tan θ 3 π (3) Compare tan θ with tan (θ – 2 ) (4) Compare tan θ with tan (θ + 18 π )+2 2 Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 Chapter five INVERSE TRIGONOMETRIC FUNCTIONS Introducing the Topic In this chapter, you and your students will learn how to compute and graph inverse trigonometric functions. The inverse is found by interchanging the range and domain of the trigonometric function. Thus, the domain of the trigonometric function becomes the range of the inverse and the range of the trigonometric function becomes the domain of the inverse. For example, the sine function has IR as its domain and -1 ≤ y ≤ 1 as its range. Therefore, the inverse of the sine function (arcsine or inverse sine) will have -1 ≤ x ≤ 1 as its domain and IR as its range. Notice, like some algebraic functions, the inverse of trigonometric function will not be a function. This is due to periodicity of the trigonometric function's domain extends to the inverse's range. Thus, one value in the inverse's domain will correspond to more than one value in the inverse's range. To make the inverse a function, you will limit the domain of the trigonometric function so it forms a one-to-one function. Limit the domain of the sine and tangent functions to -π/2 ≤ x ≤ π/2, and limit the cosine function to 0 ≤ x ≤ π . These limited domains result in one-to-one functions whose inverses are functions. The values within these limited domains are called the principal values. Notations for "inverse sine" are "arcsin" and "sin–1 ". Inverse Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 19 When asked to mentally find arcsin (1), you will typically ask yourself the question: "Sine of what angle (within the principal values) will give me 1?" The calculator does this computation very quickly for you. When graphing an inverse function by hand, you would swap the limited domain and the complete range values of the trigonometric function to form the inverse function. Once again, the calculator does this swap quickly for you. Calculator Operations To compute inverse trigonometric functions you will turn the calculator on and press + – × ÷ CL to enter computation mode and clear the home screen. The computations can be done in degrees and radians. Let's do the first one in degrees by pressing 2ndF and double touching 1 Deg. Press 2ndF (.5), press 2ndF sin-1 • 5 QUIT to exit the menu. To find sin –1 ENTER . The answer of 30° is shown below. To graph the inverse sine function, press Y= prompt. Press ENTER by pressing 2ndF sin-1 SET UP , touching B DRG, CL to access and clear the Y1 CL to clear additional Y prompts. Enter sin-1 x for Y1 X/θ/T/n ENTER . Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 4 sin-1 x. 20 Inverse Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 Method of Teaching Use Blackline Masters 5.1 and 5.2 to create overheads or handouts for demonstrating the computation and graphing of inverse trigonometric functions. Demonstrate the process in both radian and degree modes. Next, use the Blackline Master 5.3 to create a work sheet for the students. Have the students compute and graph inverse trigonometric functions. Use the topics For Discussion to supplement the worksheets. Using Blackline Master 5.2 The following problems appear on Blackline Master 5.2. To compute inverse trigonometric functions you will turn the calculator on and press + – × ÷ CL to enter computation mode. The computations can be done in degrees and radians. Let's do this one in radians by pressing 2ndF B DRG, double touching 2 Rad. Press 2ndF -1 sin (.4), press 2ndF -1 sin • 4 SET UP , touching QUIT to exit the menu. To find ENTER . The answer of .4115 radians is shown below. To find the result in terms of π press ÷ 2ndF π ENTER . So, .4115 radians is equal to .13099π. Inverse Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 21 To graph the inverse cosine function, press Y= -1 prompt. Enter cos x for Y1 by pressing 2ndF CL to access and clear the Y1 cos-1 X/θ/T/n ENTER . Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 5 cos-1 x. For Discussion Students can use the calculators to discuss the mirror imagery of a trigonometric function and its inverse across y = x, and the composition of a trigonometric function with its inverse. Demonstrate to the students the location of inverse secant, inverse cosecant, and inverse cotangent within the MATH menu. Additional Problems Compute the following: 22 1. sin-1 (-.7) = _______________ degrees 2. cos-1 (.5) = _______________ degrees 3. tan-1 (-2) = _______________ degrees 4. csc-1 (-2) = _______________ degrees 5. sin-1 (.2) = _______________ radians 6. cos-1 (-.6) = _______________ radians 7. tan-1 (3) = _______________ radians 8. cot-1 (3) = _______________ radians 9. sec-1 (1.5) = _______________ radians Inverse Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 Chapter six SOLVING TRIGONOMETRIC EQUATIONS Introducing the Topic In this chapter, you and your students will learn how to solve trigonometric equations graphically. A trigonometric equation is an equation involving a trigonometric function. Usually, trigonometric equations are solved for the principal values for each function. These values are -π/2 ≤ x ≤ π/2 for sine and tangent, and 0 ≤ x ≤ π for cosine. The other solutions will then be the integral multiples of the function's period. Some trigonometric equations will have no solutions. To solve an equation graphically, you will graph the left side of the equation as Y1 and the right side as Y2. In the case where the right side of the equation is zero, you will graph the left side as Y1 and solve for the intercepts. The point(s) of intersection will be the solutions to the equation. Calculator Operations To solve a trigonometric equation graphically, you will turn the calculator on and press Y= CL to access and clear Y1 of an old expression. Solving can be done in degrees and radians. Let's do the first one in degrees by pressing 2ndF SET UP , touching B DRG, and double touching 1 Deg. Press 2ndF 2 X/θ/T/n ) b a 2 ▼ exit the menu. To find the solutions of 3sin x – 4sin x + 1 = 0, press 3 – 4 sin X/θ/T/n + 1 QUIT to ( sin ENTER . Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 23 Set your viewing window to the principal values for sine by pressing WINDOW (–) 9 0 ENTER 9 0 ENTER 1 0 ENTER (–) 1 ENTER 1 ENTER 1 ENTER . Press GRAPH to view the graph. You will first learn how to zoom in to find a solution to the equation. You need to save this original graph and viewing window by pressing ZOOM , touching G STO, and double touching 1 StoWin. This will allow you to return to this original screen after zooming in to find the first x intercept. Set the zoom factors by pressing ZOOM , touching B FACTOR, pressing ENTER 5 5 ENTER ENTER . Press GRAPH to exit the menu and return to the graph. These larger factors will expedite the zooming in or zooming out that you desire to do. near this intercept by continuing to press the ▼ To find the left intercept, press TRACE to engage the trace and move the cursor key. Your graph should look like the following one: Now, press ZOOM , touch A ZOOM, and double touch 3 In to zoom in on the ▼ left intercept and solution of this equation. Press TRACE and to trace near and approximate the intercept. As the graph below shows, the first solution is approximately 19.4°. 24 Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 Return to the original screen by pressing ZOOM , touching H RCL, and double touching 1 RclWin. To find the right intercept, press TRACE and move the tracer near the intercept. Notice the intercept is 90°. Method of Teaching Use Blackline Masters 6.1 and 6.2 to create overheads or handouts for demonstrating the solving of trigonometric equations graphically. Demonstrate both the solving by zooming described above in Calculator Operations and the CALC feature demonstrated in Using Blackline Master 6.2. In addition, solve equations in both radian and degree modes. Next, use the Blackline Master 6.3 to create a worksheet for the students. Have the students find the solutions to the trigonometric equations through zooming and calculation in both radian and degree modes. Use the topics For Discussion to supplement the worksheets. Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 25 Using Blackline Master 6.2 To solve a trigonometric equation graphically using the CALC feature of the Sharp graphing calculator, you will turn the calculator on and press Y= CL to access and clear Y1 of an old expression. Solving can be done in degrees and radians. Let's do this one in radians by pressing 2ndF B DRG, and double touching 2 Rad. Press 2ndF QUIT to exit the menu. 2 2 ▼ To find the solutions of 2 sin x – cos x – 1 = 0, press 2 ab – cos X/θ/T/n – 1 SET UP , touching ( sin X/θ/T/n ) ENTER . Set your viewing window to the principal values for sine and cosine by pressing WINDOW (–) 2ndF π ÷ 2 ENTER 2ndF π ENTER 2ndF π ÷ 4 ENTER (–) 1 ENTER 1 ENTER 1 ENTER . Press GRAPH to view the graph. To "calculate" an intercept (solution), press 2ndF CALC and double touch 5 X_Incpt. You can calculate another intercept by pressing 2ndF CALC and double touching 5 X_Incpt again. Pressing 2ndF 26 CALC and double touching 5 X_Incpt will find the third solution. Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 To solve a trigonometric equation with functions on both the left and right sides of the equality, you will press Y= CL to access and clear Y1 of an old expression. To find the solutions of 2tan x sin x + 2 sin x = tan x + 1 you will enter the left side of the equality for Y1 and the right side for Y2. To enter the left side as Y1, press 2 tan X/θ/T/n sin X/θ/T/n ENTER . To enter the right side as Y2, press tan + X/θ/T/n 2 sin + 1 X/θ/T/n ENTER . Set your viewing window to the combined principal values by pressing WINDOW (–) 2ndF π ÷ 2 ENTER 2ndF π ENTER 2ndF π ÷ 4 ENTER (–) 2 ENTER 2 ENTER 1 ENTER . Press GRAPH to view the graph. There appears to be four solutions to the equation. To "calculate" the intersections (solutions), press TRACE , move the tracer right until it appears on the screen, and press 2ndF CALC and double touch 2 Intsct. You can calculate the next intersection by pressing 2ndF CALC and double touch 2 Intsct again. Shown below are two of the solutions (in radians) found by the CALC feature. Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 27 Calculate two more times to find the third and fourth solutions. Note that 2.3562 radians is an integral multiple of -.7854 radians. Therefore, there are three solutions -.7854, .5236, and 2.618. For Discussion Students can use the calculators to verify the additional solutions at integral multiples of the function's period. Additional Problems Solve the following equations using either the zoom or CALC feature: 1. 2cos2 x – 5cos x + 2 = 0 (in degrees) 2. tan 2 x = cot x (in degrees) 3. 2cos2 x = sin x + 1 (in degrees) 4. 3tan2 x + 4sec x + 4 = 0 (in radians) 5. cos x tan x – sin2 x = 0 (in radians) 6. 2sin x cos x + 4sin x = cos x + 2 (in radians) 28 Solving Trigonometric Equations/TRIGONOMETRY USING THE SHARP EL-9600 Chapter seven APPLICATIONS WITH RIGHT TRIANGLES Introducing the Topic The earliest applications of trigonometry involved solving right triangles. In this chapter, you and your students will explore the use of graphics calculator, particularly the SOLVER feature, in finding the solution to problems involving applications with right triangles. Calculator Operations Before using the calculator to solve problems in this chapter, set the calculator to operate in degrees by pressing 2ndF touching 1 Deg. Press 2ndF SET UP , touching B DRG, and double QUIT to exit the set up menu. Historically, the six trigonometric functions were defined in relation to a right triangle. A C C csc θ = A B cos θ = C C sec θ = B tan θ = A B B cot θ = A sin θ = C )θ B A Applications with Right Triangles/TRIGONOMETRY USING THE SHARP EL-9600 29 Given that A = 3 and C = 6, find B and θ using the SOLVER. To access the SOLVER, press 2ndF SOLVER , and press CL to delete any equation that may appear on the screen. ▼ A Type in the equation sin θ = C with the keystrokes sin ALPHA θ ALPHA = a/b ALPHA A ALPHA C . Press 2ndF SOLVER , touch A METHOD, and double touch 1 Equation to choose the Equation solver. Press ENTER and see the list of variables: Press ▼ to move the blinking cursor over the A and press 3 ENTER . Type in the value of C by pressing 6 ENTER . Solve for θ by pressing ▲ ▲ 2ndF EXE . ▼ sin. Press ▼ ▼ To use the SOLVER to find B, edit the current equation to the new equation cos A B θ = . Do this by pressing CL CL to display the current equation sin θ = C . C With the cursor over the s in sin, press cos to insert cos. Press DEL to delete to move the cursor over the A in the numerator of the fraction. Press ALPHA 30 B to insert B. Press DEL to delete A. Applications with Right Triangles/TRIGONOMETRY USING THE SHARP EL-9600 Press ENTER to see the variable list. The value for θ and C will remain as before. Press ▼ to position the cursor over the B and press 2ndF EXE to find B. Method of Teaching Use Blackline Master 7.1 to create an overhead or handout for using the SOLVER in applications with right triangles. Next, use Blackline Masters 7.2 and 7.3 to create worksheets for the students. Point out to the students that the SOLVER is an alternative to the methods of solution introduced in Chapter 5. Use the topics For Discussion to supplement the worksheets. While in the SOLVER, press 2ndF SOLVER , touch A METHOD and notice that the SOLVER utilizes three methods of solving equations. The Equation method is discussed in this chapter. The Graphic and Newton methods will be discussed in Chapter 8. Using Blackline Master 7.2 A rectangle has length 16 cm and width 8.6 cm. Find the degree measure of angle A. Determine the length of the diagonal of the rectangle. 8.6 A ( 16 Press + – × ÷ 2ndF SOLVER to access the SOLVER, and press CL to delete the equation that is on the screen. Enter the equation tan θ = A by pressing tan B Applications with Right Triangles/TRIGONOMETRY USING THE SHARP EL-9600 31 θ ALPHA = a/b ALPHA A ▼ ALPHA ALPHA B . Press ENTER to view the variable list, and enter 8.6 for A and 16 for B. (The value of θ will be listed as the one that was used in the previously entered equation.) Position the cursor over θ and press 2ndF EXE to find θ. To determine the length of the diagonal of the rectangle, return to the equation by pressing CL CL and edit the current equation to read sin θ = A . C The value of θ should be as before, so enter 8.6 for A and solve for C. For Discussion Students can edit a current equation in the SOLVER to a new equation instead of clearing the current equation and typing in the new equation. 32 Applications with Right Triangles/TRIGONOMETRY USING THE SHARP EL-9600 Additional Problems Problems 1-5 use the following figure: U β θ ) C A T R S ) α B 1. If θ=42° and U=8, find T. 2. If α=25° and U=16, find S+T. 3. If R=10 and S=8, find the degree measure of α. 4. If S=2, R=3.5, and U=4.8, find T. 5. Is it possible for the distance between points C and D to be 4 if S=2 and R=3.5? Explain why or why not. 6. A farmer wants to put a fence around the perimeter of a flat field that is in the shape of a right triangle. One angle of the triangle is 42° and the length of the side opposite this angle is 60 meters. Find the length of fencing that is required to enclose the field. Applications with Right Triangles/TRIGONOMETRY USING THE SHARP EL-9600 33 Chapter eight LAW OF SINES AND LAW OF COSINES Introducing the Topic Chapter 7 considered techniques of finding the solution to problems involving right triangles. This chapter explores using the calculator with the Law of Sines and Law of Cosines. These two laws relate the angles and lengths of the sides of any triangle and have been used for many centuries in solving practical problems such as those that occur in map making, navigation, and surveying. Calculator Operations Before using the calculator to solve problems in this chapter, set the calculator to operate in degrees by pressing 2ndF touching 1 Deg. Press 2ndF SET UP , touching B DRG, and double QUIT to exit the set up screen. The Law of Sines states that the ratio of the lengths of the sides of a triangle to each other is the same as the ratio of the sines of the angles opposite the sides. That is, if A, B, and C are the measures of the three angles of a triangle and X, Y, ( and Z are respectively the lengths of the three sides opposite those angles, then: Z B A X ( C sin A = sin B = sin C Z Y X ( Y 34 Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 You will find the SOLVER very useful when using the Law of Sines or Law of Cosines. To access the SOLVER, press 2ndF SOLVER , and press CL to delete any equation that may appear on the screen. To use the SOLVER with the Law of Sines, you can enter any of the following three equivalent statements: sin A = sin C X Z sin A = sin B X Y sin B = sin C Y Z B ALPHA ALPHA A ALPHA X ▼ sin ▼ a/b ▼ Since you can label the sides and the angles of any triangle to correspond to one of these three statements, choose one, say sin A = sin B , and enter it with X Y the keystrokes: ALPHA = a/b sin ALPHA Y Since you will be using this formula many times as you work through this chapter, store it into memory by pressing 2ndF SOLVER , touching C SAVE, pressing ENTER , and typing in a name of your choice, say LAWSINES by pressing L A W S I N E S . Note that the cursor is automatically set to ALPHA, so you can just type letters. Press ENTER to store the formula. The equation you have stored will reappear on the screen. Press 2ndF SOLVER , touch A METHOD, double touch 1 Equation to choose the Equation Solver. Press ENTER , and you see the list of variables for your equation. You may see values other than zero. y Consider the triangle shown on the right. What is the value of y? What is the value of z? A=95° z B=48° x = 15 With the blinking cursor over the A, type in 95 and press ENTER . Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 35 Next type 15 for x, the length of the side opposite the 95°, and press ENTER . Enter 48 for B in the same manner. Solve for y by positioning the blinking cursor over the y and press 2ndF EXE . The y turns out to be about 11.19. Use the fact that the sum of the angles in any triangle is 180° to find 37° as the measure of the angle opposite side z. You can now apply methods in Chapter 7 to edit the current equation to one involving a known side and side z, say sin A = sin C , or you can work with the current equation using B = 37 and X Z y = z. In either case, you should find z = 9.06. This problem could also be solved using the Graphic Solver. Press 2ndF SOLVER , touch A METHOD, double touch 3 Graphic to choose the Graphic Solver. Enter the known quantities for the first part of this example: angle A measures 95°, side x is of length 15, and angle B measures 48°. (To start the problem from the beginning, enter 0 for y). Use ▼ to position the cursor over the unknown quantity, y. To begin the solution process press 2ndF EXE and you will see: You must enter the range values for the unknown (from BEGIN to END) within which you expect to find the solution. Let the BEGIN = 5 by pressing 5 ENTER . Enter a value for END that you do not expect y to exceed, say 25, by pressing 2 5 ENTER . Press 2ndF EXE and an autoscaled plot will be drawn showing graphs of the left and right sides of the current equation. The calculator will now display a blinking box in the upper right-hand corner of the viewing screen while it looks for the intersection of the two graphs. If one is found, the solution is displayed at the bottom of the screen. 36 Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 Consider another example. Suppose the length of the side of a triangle opposite a 65° angle is 4 and another side is of length 12. Find the measure of the angle opposite the side of length 12. Press CL to view the variable list. Enter A = 65, x = 4, and y = 12. Position the cursor on B and press 2ndF EXE to solve for the unknown angle. Since any angle of a triangle is between 0° and 180°, enter these values for BEGIN and END. Press 2ndF EXE and you will see that the graphs of the left- and right-hand A sides of the equation sin = sin B do not intersect. x y The message “No solution” in window is then displayed at the bottom of the screen. If you get this message when using the Graphic Solver, you should enter another BEGIN and/or END within which the Graphic Solver will search for solutions. In this case, however, you have included from BEGIN to END all possible values for the measure of angle B. Thus, there is no triangle that could have the indicated measures. You can algebraically confirm this fact by noticing that sin B = (Y sin A)/X= 3 sin 65° = 2.719 >1 which is of course, impossible. The Law of Cosines is helpful in solving problems when the lengths of two sides and the angle between those sides is known or when the lengths of all three sides are known. This law says that the square of the length of any one side of a triangle equals the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle between those sides. That is, if x, y, and z are the lengths of the three sides of a triangle and θ is the angle between the two sides of lengths x and y, then: z2 = x2 + y2 – 2xy cos θ Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 37 Access the SOLVER with 2ndF SOLVER (or press CL twice if you are still in the SOLVER). Press CL to delete the current equation. Return to the Equation Solver by pressing 2ndF SOLVER , touch A METHOD, and double touch 1 Equation. Next enter the Law of Cosines with the keystrokes: Z x2 ALPHA ALPHA X × ALPHA 2 = ALPHA ALPHA Y × X cos x2 + ALPHA ALPHA Y x2 – θ Store this equation in the calculator using the method outlined to store the Law of Sines. Suppose you are given a triangle in which two sides of lengths 13 cm and 8 cm enclose a 50° angle. Find the length of the third side of the triangle. To find the solution using the Law of Cosines, press ENTER to view the list of variables: Z, X, Y, and θ. Press ▼ to position the cursor on X and enter 13. Next, enter 8 for Y and 50 for θ. Press ▲ to move the cursor over Z and press 0 ENTER . Press 2ndF EXE to solve for Z. Notice that the Equation Solver has automatically changed to the Newton Solver. The calculator has chosen this method of solution since the equation is such that a simple rearrangement of the variables will not produce a value for the unknown. The Newton method uses iterations to approximate the root of the equation. You should supply an initial estimate of the solution in the START position (0 is the default value) and a step size STEP (.001 is the default value) that the SOLVER uses to determine successive estimates. Press 2ndF 38 EXE to find the solution. Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 (If you receive an error message while using the Newton Solver, set the START value at your estimate of the solution and try slightly increasing the STEP value.) The method is continued until the left side and right side of the equation are within a defined tolerance. Since the left and right sides of the equation are displayed as equal values, and the L – R = 0, the NEWTON SOLVER has found a solution that is correct to the limits of the calculator’s accuracy. Method of Teaching Use Blackline Master 8.1 to create an overhead or handout for using the Graphic Solver with the Law of Sines. Use Blackline Master 8.2 to create an overhead or handout for using the Newton Solver with the Law of Cosines. Demonstrate the procedure to store each of these equations in the calculator memory for later recall. Next, use Blackline Master 8.3 to create a worksheet for the students. Remind them that some problems may be solved by a variety of methods. If information is not available to obtain a good estimate of START in the Newton Solver, students may find it more convenient to work with the Graphic Solver. Use the topics For Discussion to supplement the worksheets. Retrieving (loading) and renaming equations that have been stored in the calculator is discussed in the owner's manual. Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 39 Using Blackline Master 8.3 The problems discussed in the Calculator Operations section are presented on Blackline Masters 8.1 and 8.2. Problem 1 on Blackline Master 8.3 involves the angles of elevation of the sun. Remind students that the angle of elevation is the angle, measured upward from the horizontal, to the line of sight of an object. The Graphic Solver provides an added exploration when finding the solution to this problem. In the variable list for the Law of Sines, enter 60 for A, 50 for X, and 40 for Y. Solve for B using BEGIN= 0 and END= 180 to obtain B = 43.85°. Notice, however, that two intersections are visible. Press 2ndF CALC and the Graphic Solver will find the other intersection at B =136.15°. Have students discuss why this is not a possible value for B. Once B is found, 90 – θ = 180 – (60 + 43.85) = 76.15 yielding θ = 13.85°. For Discussion Students can discuss how to retrieve and rename the stored equations for the Law of Sines and the Law of Cosines. Additional Problems T Problems 1-3 refer to the following situation: 40 ft A tower on the top of a hill is 40 feet tall. d The angle of elevation from point B at the base of the hill to the top of the tower (point h T) is 45°. The angle of elevation from point A, 800 feet from the base of the hill, to 22° A 800 ft B 45° point T is 22°. 1. Find the distance d from point A to the top of the tower. 2. A cable is stretched taunt between points B and T. Find the length of the cable. 3. Find the height h of the hill. 4. The angle of elevation of the top of a flagpole from point A is 22°. The angle of depression of helicopter ) 25° 1200 ft the top of the flagpole from a helicopter hovering R 1200 feet above point A is 25°. Find the distance from the helicopter to the top of the flagpole. 40 ) 22° A Law of Sines and Law of Cosines/TRIGONOMETRY USING THE SHARP EL-9600 Chapter nine AREA OF A TRIANGLE Introducing the Topic In this chapter, you and your students will explore three formulas and the use of the graphing calculator, particularly the SOLVER, in finding the area of a triangle. Calculator Operations Before using the calculator to solve problems in this chapter, set the calculator to operate in degrees with 2ndF 1 Deg. Press 2ndF SET UP , touch B DRG, and double touch QUIT to exit the set up screen. Depending on the information given in the problem situation, there are three formulas you can use to find the area of any triangle: 1. The area of a triangle is one-half the product of the base and the height where the height is the length of the line segment drawn from a vertex of the triangle perpendicular to the opposite side called the base. height area = 1 base•height 2 base Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 41 2. The area of a triangle is one-half the product of the lengths of two sides of the triangle and the measure of the angle included between those sides. b 1 area = 2 bc sin θ θ c 3. In the second half of the first century A.D., Heron of Alexandria stated the following formula for the area of any triangle in terms of the lengths of its sides. It is known as Heron’s formula (or Hero’s formula): Y X area = √ [S(S-X)(S-Y)(S-Z)] where S = 1 (X +Y +Z ). 2 Z 1 The first formula for the area of a triangle, area = 2 base • height, can be typed as A = 1 B•H. Note that the multiplication key × must be pressed between the 2 base and height to separate the variable names. To access the SOLVER, press 2ndF SOLVER , and press CL to delete any equation on the screen. a/b 1 Press 2ndF 2 ▼ = ▼ Type in the equation A = 1 B • H with the keystrokes: ALPHA 2 ALPHA B × ALPHA H . A ALPHA SOLVER , touch A METHOD, and double touch 1 Equation to choose the Equation solver. Press ENTER to see the list of variables for your equation: 42 Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 To find the area of a triangular garden plot with base = 4 meters and height = 8.35 meters, use ▼ to move the blinking cursor over B and type in 4. Type in 8.35 for H and press ENTER . Press ▲ twice to move the prompt to the A variable. Press 2ndF EXE to find the area, 16.7 square meters. (You may wish to save this formula for later recall using the methods described in Chapter 7.) The second formula for the area of a triangle can be typed in as A = 1 B•C•sin θ. Press CL three times to return to and delete the current equation. 2 ALPHA ▼ ▼ 1 Type in the equation A = 2 B•C•sin θ with the keystrokes: ALPHA A ALPHA = a/b 1 2 ALPHA B × ALPHA C × sin θ . Consider the triangle below where sides of lengths 4.8 and 5.73 enclose a 47° angle. The height of this triangle is not known, but you do have 1 the necessary information to use the formula A = B•C•sin θ. 2 5.73 47° ( 4.8 Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 43 Press ENTER to access the variable list, enter 5.73 for B, 4.8 for C (or viceversa), and 47 for θ. Move the cursor to the A variable. Press 2ndF EXE to solve for the area. Method of Teaching Use Blackline Master 9.1 to create an overhead or handout for using the SOLVER in applications with areas of triangles. Remind the students to store each of the formulas under a name of their choice before deleting the current formula and entering a new one. Next, use Blackline Masters 9.2, 9.3, and 9.4 to create worksheets for the students. Encourage different methods of solution and use of both the Equation solver and Graphic solver. Use the topics For Discussion to supplement the worksheets. Using Blackline Master 9.2 The second problem presented under the CALCULATOR OPERATIONS section, also provides a nice exercise with the Graphic solver. Press 2ndF SOLVER , touch A METHOD, and double touch 3 Graphic to choose the Graphic solver. To start the problem from the beginning, set A to 0. Move the cursor to the A variable and press 2ndF EXE to see the Graphic solver variable range screen. Recall that the Graphic solver searches for the intersection of the left and right hand sides of your equation from BEGIN to END. Since area is always nonnegative, 0 could be used for BEGIN. Enter END a value that you do not expect area to exceed, say 25. (A little bit of estimation helps here! If your value for END is too small, increase it and try again.) 44 Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 Press 2ndF EXE and an autoscaled graph of the two sides of the equation is drawn and the solution is obtained. Another formula for area of a triangle is Heron's formula. Enter Heron’s formula, A = √ [S(S-X)(S-Y)(S-Z)], press CL three times, and type the following: ALPHA A ALPHA = 2ndF √ ALPHA S ( ALPHA S – ALPHA X ) ( ALPHA S – ALPHA Y ) ( ALPHA S – ALPHA Z ) . (Note that the square root symbol extends over the entire right hand side of the equation, automatically grouping the expression under it.) 1 You could enter the semiperimeter, S = 2 (X + Y + Z), in a separate formula in the SOLVER, but it is probably easier to find the value of S in the computation mode and then use Heron’s formula in the SOLVER. Press ENTER to view the variable list. For Discussion Students can discuss the steps to enter the distance formula in the SOLVER. Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 45 Additional Problems Problems 1-3 refer to the figure below: C θ υ b a h A β ) α c D d ( B 1. If α = 60°, c = 7 and b = 10, find the area of triangle ACD. 2. If a = 2.5, d = 1.5 and h = 2, find the area of triangle DCB. 3. If h = 14.68, c = 16, and d = 10, find the area of triangle ACB. 4. Find the area of a triangle with vertices at (-1, 5), (1, -1) and (4, 2). 5. Suppose you are told that a triangle with two sides of length 12 and 15 has an area equal to 200 and you are asked to find the measure of the angle between the two given sides. What would you find? Why did you find it? 46 Area of a Triangle/TRIGONOMETRY USING THE SHARP EL-9600 Chapter ten AREA OF A CIRCULAR SECTOR Introducing the Topic Let’s explore one of the many uses for angles measured in radians, and that is determining the area of a sector of a circle. Calculator Operations Before using the calculator to solve problems in this chapter, set the calculator to operate in radians by pressing 2ndF touching 2 Rad. Press 2ndF SET UP , touching B DRG, and double QUIT to exit the set up screen. A portion of a curve is called an arc. When the curve is a circle, a central angle is an angle whose vertex is at the center of the circle. The central angle is said to intercept, or subtend, the arc. Area of a Circular Sector/TRIGONOMETRY USING THE SHARP EL-9600 47 The portion of the circle and its interior that is formed by the two radii enclosing a central angle and an arc is called a sector of the circle. The formula for the area of a circular sector in terms of the radius r of the circle and the 1 central angle θ, measured in radians, is area = r 2θ. 2 To enter this formula in the calculator, access the SOLVER by pressing + – × ÷ 2ndF SOLVER . Press CL to delete any equation that appears on the screen. a/b 2 ▼ Type in the equation A= 12 R2θ by pressing ALPHA ALPHA R 2 x ALPHA A ALPHA = 1 θ . Consider this problem situation. A lawn sprinkler is set so that the region of the lawn that is watered is approximately a sector of a circle of radius 6 feet. If the central angle of the sector is π radians, find the area of the region that is 5 watered by the sprinkler. 6 ft π 5 48 6 ft Area of a Circular Sector/TRIGONOMETRY USING THE SHARP EL-9600 To find the area using the Equation solver, first press 2ndF SOLVER , touch A METHOD, and double touch 1 Equation. Press ENTER and see the list of variables: A, R and θ. Use ▼ to place the cursor over R, type in 6, and press ENTER . Type in θ by pressing 2ndF π ÷ 5 ENTER . Press ▲ twice to place the cursor on the A variable. Press 2ndF EXE to find the area in square feet. To find the area using the Newton solver, first press 2ndF SOLVER , touch A METHOD, and double touch 2 Newton. Reset A to 0 to start the problem from the beginning. The values of R and θ are correct for this problem, so use ▲ to place the cursor over the A variable and press 2ndF EXE to view the Newton solver screen. Since area is never negative, you could set START=0. Leave the STEP size as .001 and remember that you should change it to a slightly larger value if an error message results. Press 2ndF EXE to find the area in square feet. Area of a Circular Sector/TRIGONOMETRY USING THE SHARP EL-9600 49 Method of Teaching Use Blackline Masters 10.1 and 10.2 to create overheads or handouts for using the SOLVER in applications with the area of a circular sector. Demonstrate the steps to prepare the calculator for use with the SOLVER, and emphasize the fact that the central angle in the formula for the area of a sector of a circle must be in radians, not degrees. Encourage students to give units with all answers. Next, use Blackline Master 10.3 to create a worksheet for finding the area of a circular sector. Encourage students to explore all three techniques (Equation, Newton, and Graphic) available in the SOLVER so they will have experience with other methods if the one they are attempting to use fails. Use the topics For Discussion provided to supplement the worksheets. Using Blackline Master 10.2 To find the area using the Graphic Solver, press 2ndF SOLVER , touch A METHOD, and double touch 3 Graphic. Reset A to 0 to start the problem from the beginning. Place the cursor on the A variable and press 2ndF EXE . Recall that BEGIN and END are the lower and upper values between which you feel the solution will lie. Enter a 20 for END so that the intersection of the two graphs is visible on the screen. Press 2ndF 50 EXE to find the area. Area of a Circular Sector/TRIGONOMETRY USING THE SHARP EL-9600 For Discussion Students can discuss the advantages and/or disadvantages of using the three calculator solvers: Equation solver, Newton solver, and Graphic solver in various problem situations. Additional Problems 1. A circle has radius 6 cm. Find the area of a sector of this circle whose arc subtends an angle of measure 3π radians. 4 2. A circular garden has diameter 8 feet. Broccoli is to be planted in a sector of this garden whose arc subtends an angle of 50°. Find the area of the broccoli sector. 3. If the area of a sector of a circle is 7.52 square inches and the radius of the circle is 2.4 inches, find the measure of the angle subtended by the arc of the sector. (Give units with your answer.) Area of a Circular Sector/TRIGONOMETRY USING THE SHARP EL-9600 51 CONTENTS OF REPRODUCIBLE BLACKLINE MASTERS Use these reproducible Blackline Masters to create handouts, overhead transparencies and activity worksheets. GRAPHS OF TRIGONOMETRIC FUNCTIONS BLACKLINE MASTERS 1.1 - 1.4 53 - 56 EVALUATING TRIGONOMETRIC FUNCTIONS BLACKLINE MASTERS 2.1 - 2.4 57 - 60 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES BLACKLINE MASTERS 3.1 - 3.5 61 - 65 AMPLITUDE, PERIOD AND PHASE SHIFT BLACKLINE MASTERS 4.1 - 4.3 66 - 68 INVERSE TRIGONOMETRIC FUNCTIONS BLACKLINE MASTERS 5.1 - 5.3 68 - 71 SOLVING TRIGONOMETRIC EQUATIONS BLACKLINE MASTERS 6.1 - 6.3 72 - 74 APPLICATIONS WITH RIGHT TRIANGLES BLACKLINE MASTERS 7.1 - 7.3 75 - 77 LAW OF SINES AND LAW OF COSINES BLACKLINE MASTERS 8.1 - 8.3 78 - 80 AREA OF A TRIANGLE BLACKLINE MASTERS 9.1 - 9.4 81 - 84 AREA OF A CIRCULAR SECTOR BLACKLINE MASTERS 10.1 - 10.3 KEYPAD FOR THE SHARP EL-9600 52 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 85 - 87 88 NAME _____________________________________________________ CLASS __________ DATE __________ 1.1 GRAPHS OF TRIGONOMETRIC FUNCTIONS Steps for graphing the sine trigonometric function 1. Turn the calculator on and press Y= . 2. Press CL to remove an old Y1 expression. 3. Press ▼ CL to clear additional expressions. 4. To set up the calculator press 2ndF SET UP , touch B DRG, double touch 2 Rad, touch E COORD, double touch 1 Rect, touch G EDITOR, and double touch 1 Equation. QUIT again to return to the Y prompts. 6. Enter the sine function ( y = sin x ) for Y1, by pressing sin X/θ/T/n . 7. Enter the viewing window range by pressing ZOOM , touching E TRIG, and double touching 1 sin x. Steps for graphing the cosecant trigonometric function. 1. Press Y= CL to remove an old Y1 expression. 2. Enter the cosecant function for Y1 by pressing MATH , touch A CALC, double touch 10 csc, and press X/θ/T/n . 3. Continue to use the sine function's built-in viewing range. 4. Press GRAPH to view the graph. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 53 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 5. Press 2ndF NAME _____________________________________________________ CLASS __________ DATE __________ 1.2 GRAPHS OF TRIGONOMETRIC FUNCTIONS Activity 1 Graph the cosine function. 1. Turn the calculator on and press Y= . 2. Press CL to remove an old Y1 expression. 3. Press ▼ CL to clear additional expressions. 4. To set up the calculator press 2ndF SET UP , touch B DRG, double touch 2 Rad, touch E COORD, double touch 1 Rect, touch G EDITOR, and double touch 1 Equation. 5. Press 2ndF QUIT again to return to the Y prompts. Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 6. Enter the cosine function ( y = cos x ) for Y1, by pressing cos X/θ/T/n . 7. Enter the viewing window range by pressing ZOOM , touching E TRIG, and double touching 2 cos x. 8. Sketch your graph in the box. 54 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 1.3 GRAPHS OF TRIGONOMETRIC FUNCTIONS Activity 2 Graph the tangent function. 1. Turn the calculator on and press Y= . 2. Press CL to remove an old Y1 expression. 3. Press ▼ CL to clear additional expressions. 4. To set up the calculator press 2ndF SET UP , touch B DRG, double touch 2 Rad, touch E COORD, double touch 1 Rect, touch G EDITOR, and double touch 1 Equation. QUIT again to return to the Y prompts. 6. Enter the cosine function ( y = tan x ) for Y1, by pressing tan X/θ/T/n . 7. Enter the viewing window range by pressing ZOOM , touching E TRIG, and double touching 3 tan x. 8. Sketch your graph in the box. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 55 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 5. Press 2ndF NAME _____________________________________________________ CLASS __________ DATE __________ 1.4 GRAPHS OF TRIGONOMETRIC FUNCTIONS Activity 3 Graph the secant function. 1. Press Y= CL to remove an old Y1 expression. 2. Enter the secant function for Y1 by pressing MATH , touch A CALC double touch 09 sec, and press X/θ/T/n . 3. Use the cosine function's built-in viewing range by pressing ZOOM , Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. touching E TRIG, and double touching 2 cos x. Activity 4 Graph the cotangent function. Try graphing the cotangent function without looking at the steps provided in Activities 1-3. Use the built-in viewing window range for tangent and sketch your graph in the box below. 56 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 2.1 EVALUATING TRIGONOMETRIC FUNCTIONS Steps for evaluating sin 45° directly 1. Turn the calculator on and press + – × ÷ to enter computation mode. 2. Press CL to clear the viewing window of any old expressions. 3. Set the calculator to operate in degrees by pressing 2ndF B DRG, and double touch 1 Deg. Press 2ndF SET UP touch QUIT to exit the menu. 4. Enter and evaluate the expression by pressing sin 4 5 ENTER . Steps for approximating sin ( π 3 ) graphically 2. Enter the function sin x by pressing sin radian mode by pressing 2nd F 2 Rad. Press 2nd F X/θ/T/n . Set the calculator to SET UP , touch B DRG, and double touch QUIT to exit the menu. 3. View the graph by pressing ZOOM , touching E TRIG, double touching 1 sin x. 4. Now, engage the trace by pressing TRACE and tracing near x = π = 1.047. 3 5. With the cursor located near x = 1.047, you can zoom in to find a better approximation for the y value. Press ZOOM , touch A ZOOM, and double touch 3 In to zoom in on the cursor. 6. Once again, trace over near x = 1.047. Zoom in one more time to see an even better approximation. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 57 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 1. Press Y= followed by CL to clear an old expression for Y1. NAME _____________________________________________________ CLASS __________ DATE __________ 2.2 EVALUATING TRIGONOMETRIC FUNCTIONS Steps for evaluating sec ( π 4 ) directly in degree mode 1. Press + – × ÷ to enter computation mode. 2. Press 2ndF SET UP to verify the calculator is in degree mode. If it is not, change it to degree mode. Press 2ndF QUIT to exit the menu. double touching 09 sec, pressing 2ndF π a/b 4 ▼ 3. Enter and evaluate the expression by pressing MATH , touching A CALC, MATH , touching E ANGLE, double touching 4 r, and pressing ENTER . Steps for approximating tan 20° graphically Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 1. Press Y= and press CL to remove an old expression for Y1. 2. Press 2ndF SET UP to verify the calculator is in degree mode. If it is not, change it. Press 2ndF QUIT to exit the menu. 3. Enter the function tan x for Y1 by pressing tan X/θ/T/n . 4. Instead of graphing the function in the default viewing window for tangent, press WINDOW adjust the Xmin and Xmax to 10° and 30° (value of interest ± 10° ) respectively to start out closer to the value of interest. Set the Xscl = 1. Press ZOOM , touch A ZOOM, and double touch 1 Auto to view the graph. 5. Now, engage the trace by pressing TRACE . Notice the tracer appears at x = 20° since it is the middle of the x values. 58 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 2.3 EVALUATING TRIGONOMETRIC FUNCTIONS Direct evaluation of trigonometric functions in degrees 1. sin 80° 2. cos (-10°) 3. tan 70° 4. csc 55° 5. sec 22° 6. cot 40° Activity 2 Direct evaluation of trigonometric functions in radians π 7. sin (– 5 ) 7π 8. cos ( 6 ) π 9. tan ( 4 ) 5π 10. csc ( 4 ) 11. sec ( 2π ) 3 12. cot (– π ) 7 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 59 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. Activity 1 NAME _____________________________________________________ CLASS __________ DATE __________ 2.4 EVALUATING TRIGONOMETRIC FUNCTIONS Activity 3 13. sin 18° 14. cos 33° 15. tan (-134°) 16. csc 85° 17. sec 145° 18. cot 235° Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. Activity 4 60 Graphical approximation of trigonometric functions in degrees Graphical approximation of trigonometric functions in radians 4π 19. sin ( 5 ) 20. 6π cos ( 7 ) 21. 3π tan ( 4 ) π 22. csc ( 9 ) 23. sec ( 3π ) 8 24. 5π cot ( 11 ) Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 3.1 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES 1. Verify the double angle formula sin 2θ = 2 sin θ cos θ, by graphing Y1 = sin (2x) and Y2 = 2 sin x cos x and show that they are in fact the same graph. 2. Turn the calculator on and press Y= . Press CL to clear Y1 of an old expression. Press ▼ CL to clear additional expressions. 3. Make sure the calculator is in radian mode by pressing 2ndF touch B DRG, and double touch 2 Rad. Press 2ndF 4. Now enter sin (2x) for Y1 by pressing sin and the 2 sin x cos x for Y2 by pressing 2 ( sin 2 SET UP , QUIT to exit the menu. X/θ/T/n X/θ/T/n ) cos ENTER X/θ/T/n ENTER . ➧ 5. Set your viewing window by pressing WINDOW EZ , touch 5, touch on the screen, touch 6 -2π<X,<2π, touch on the screen, and touch 5 -5<Y<5. 61 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. ➧ Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 3.2 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES θ Verify the half angle formula cos ( 2 ) = ± √( (1 + cos θ) ) : 2 1. To verify the formula, you will graph Y1 = cos x and Y2 = ±√( (1 + cos x) ) 2 2 and show that they are in fact th same graph. 2. In this problem you will need to limit your domain or x values to intervals where cosine is either negative or positive and enter the appropriate radical for Y2. 3. To do this you will turn the calculator on and press Y= . Press CL to clear Y1 of an old expression. Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. x 4. Now, enter the cos 2 for Y1 by pressing cos X/θ/T/n a/b 2 ENTER x) ) for Y2 (limiting to where cosine is positive) by and the √ ((1 + cos 2 pressing CL 2ndF √ a/b 1 + cos X/θ/T/n ▼ 2 ENTER . x 5. Cos ( 2 ) will be positive for a viewing window of 0 < x < π . Enter this viewing window by pressing WINDOW 0 ENTER 2ndF π ENTER 2ndF π ÷ 2 ENTER (–) 3 ENTER 3 ENTER 1 ENTER . 6. Press GRAPH to view the graphs. Notice they appear as one graph. 7. To verify the formula for where cosine is negative, press Y= ▼ to access the Y2 prompt and press (–) to insert a negative in front of the radical. x Cos ( 2 ) will be negative for a viewing window of π< x < 3π. Enter this viewing window by pressing WINDOW 2ndF π ENTER 3 2ndF π ENTER . Press GRAPH to view the graphs. 62 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 3.3 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES Activity 1 Verify the half-angle formula sin ( θ ) = ± √( (1 – cos θ) ): 2 2 1. Press Y= CL to clear Y1 of an old expression. Press ENTER CL to clear additional expressions. 2. Make sure the calculator is in radian mode by pressing 2ndF touching B DRG, and double touching 2 Rad. Press 2ndF SET UP , QUIT to exit the set-up menu. x 3. Enter the sin ( 2 ) for Y1 by pressing sin X/θ/T/n a/b 2 ENTER . 4. Enter the √ ( (1 – cos x) ) for Y2 (limiting domain to an interval where sine 2 is positive) by pressing 2ndF √ a/b 1 – cos X/θ/T/n ▼ 2 x 5. Sin ( 2 ) is positive for 0 < x < 2π. Enter this range by pressing WINDOW 0 ENTER 2 2ndF π ENTER 2ndF π ÷ 2 ENTER (–) 2 ENTER 2 ENTER 1 ENTER . 6. Check to see if they result in the same graph by pressing GRAPH . Sketch what you see in the box below: Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 63 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. ENTER . NAME _____________________________________________________ CLASS __________ DATE __________ 3.4 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES Activity 2 Verify the formula from Activity 1 for intervals where sine is negative: 1. Press Y= ▼ to access the Y2 prompt. 2. Press (–) to insert a negative in front of the radical. x 3. Sin ( 2 ) will be negative for 2π < x < 4π. Enter this range by pressing WINDOW 2 2ndF π ENTER 4 2ndF π ENTER . 4. Check to see if they result in the same graph by pressing GRAPH . Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 5. Sketch the graph in the box below. 64 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 3.5 GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES Activity 3 Verify the identity tan2 θ + 1 = sec 2 θ: 1. Access and clear the Y1 prompt. 2. Enter the tan2 x + 1 for Y1. 3. Enter the sec2 x for Y2. 4. Set your viewing window. 5. Check to see if they result in the same graph. Activity 4 Verify the following formulas and identities using the steps provided in Activities 1 through 3: 1. cos 2θ = 2cos2 θ – 1 2. csc2 θ – cot2 θ = 1 3. tan θ csc θ = sec θ Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 65 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 6. Sketch the graph in the box below. NAME _____________________________________________________ CLASS __________ DATE __________ 4.1 AMPLITUDE Investigate changes in amplitude with the SHIFT/CHANGE feature. 1. Press 2ndF SHIFT/CHANGE , touch B CHANGE, and double touch 5 Y=sin X. 2. Double touch the screen at Y = 3, above the box with the arrows. Notice the new function Y = 3 sin X is displayed. 3. Press ENTER to place the new function in the list. 4. Double touch the screen at Y = -2, below the box with the arrows. Notice the new function Y = -2 sin X is displayed. 5. Press ENTER to place the new function in the list. Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 6. Touch the top of the right side of the screen and the list of three functions will be displayed. 7. The function Y = sin X is highlighted on the right side of the screen, and the graph is darkened on the left side of the screen. 8. Press ▼ to move from one graph to the next. Observe and note the changes in sin X when you change the amplitude. _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 66 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 4.2 PERIOD AND PHASE SHIFT Investigate the change in a trigonometric function by multiplying the argument θ by 2. 1. Press Y= CL to access and clear Y1 of an old expression. 2. Make sure the calculator is in radian mode by pressing 2ndF touch B DRG, and double touch 2 Rad. Press 2ndF SET UP , QUIT to exit the menu. 3. Now, enter sin x for Y1 by pressing sin X/θ/T/n ENTER and the sin (2x) for Y2 by pressing CL sin ( 2 X/θ/T/n ) ENTER . 4. Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 1 sin x. Note the change in the second graph versus the first Investigate the change in a trigonometric function by subtracting 3π to the argument θ. 2 1. Press 2ndF SHIFT/CHANGE , touch A SHIFT, and double touch 6 Y = sin X. Double touch the screen at X = 3π/2 to the right of the box with the arrows. Notice the new function Y = sin (X - 3π/2) is displayed. Press ENTER to place the new function in the list. 2. Double touch the screen at Y = -2, below the box with the arrows. Notice the new function Y = sin (X – 3π/2) -2 is displayed. Press ENTER to place the new function in the list. Touch the top of the right side of the screen and the list of three functions will be displayed. The function Y = sin X is highlighted on the right side of the screen, and the graph is darkened on the left side of the screen. Press ▼ to move from one graph to the next. Observe and note the changes in sin X when you change the phase shift. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 67 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. graph. Engage the trace to show the period has changed from 2π to π. NAME _____________________________________________________ CLASS __________ DATE __________ 4.3 AMPLITUDE, PERIOD, AND PHASE SHIFT Activities Enter the first graph for Y1 with the second graph for Y2 and note the change(s). Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. COMPARE 1. cos x with 2cos x ____________________________ 1 2. cos x with ( 2 )cos x ____________________________ 3. cos x with cos 2x ____________________________ x 4. cos x with cos ( 2 ) ____________________________ 5. cos x with cos(x – π ) 6 ____________________________ 6. cos x with cos(x + π ) 6 ____________________________ 7. cos x with 3cos(x + π ) 6 ____________________________ 8. tan x with 2tan( x ) 3 ____________________________ x π 9. cot x with cot( 2 – 3 ) ____________________________ 10. sec x with 2sec(3x) ____________________________ 11. csc x with ( 68 CHANGE(S) 1 )csc(x – π ) 2 6 ____________________________ Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 5.1 INVERSE TRIGONOMETRIC FUNCTIONS To compute the inverse sine function in degrees: 1. Turn the calculator on and press + – × ÷ CL to enter computation mode. 2. Change to degree mode by pressing 2ndF double touching 1 Deg. Press 2ndF 3. Find sin–1 (.5) by pressing 2ndF sin-1 SET UP , touching B DRG, and QUIT to exit the menu. • 5 ENTER . The answer of 30° To graph the inverse sine function in degrees: 1. Press Y= CL to access and clear the Y1 expression. 2. Press ENTER CL to clear additional Y expressions. 3. Enter sin-1 x for Y1 by pressing 2ndF sin-1 X/θ/T/n ENTER . 4. Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 4 sin-1 x. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 69 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. is shown below. NAME _____________________________________________________ CLASS __________ DATE __________ 5.2 INVERSE TRIGONOMETRIC FUNCTIONS To compute the inverse sine function in radians: 1. Turn the calculator on and press + – × ÷ CL to access and clear the home screen. 2. Change to radian mode by pressing 2ndF double touching 2 Rad. Press 2ndF 3. Find sin-1 (.4), press 2ndF sin-1 • SET UP , touching B DRG, QUIT to exit the menu. 4 ENTER . The answer of .4115 radians is shown below. To find the result in terms of π press ÷ Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. π 2ndF ENTER . To graph the inverse cosine function in radians: 1. Press Y= CL to access and clear the Y1 prompt. 2. Enter cos -1 x for Y1 by pressing 2ndF cos-1 X/θ/T/n ENTER . 3. Set your viewing window by pressing ZOOM , touching E TRIG, and double touching 5 cos-1 x. 70 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 5.3 INVERSE TRIGONOMETRIC FUNCTIONS Activities 1. sin-1 .8 = _______________ degrees 2. cos-1 (–.6) = _______________ degrees 3. tan-1 2 = _______________ degrees 4. sec-1 (-1.3) = _______________ radians 5. csc-1 (1.7) = _______________ radians 6. cot-1 (-1) = _______________ radians Graph the following. 7. y = tan-1 x 8. y = cot-1 x Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 71 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. Compute the following. NAME _____________________________________________________ CLASS __________ DATE __________ 6.1 SOLVING TRIGONOMETRIC EQUATIONS To solve a trigonometric equation graphically using the zoom feature: 1. Turn the calculator on and press Y= CL to access and clear Y1 of an old expression. Set the calculator to degree mode by pressing 2ndF touching B DRG, and double touching 1 Deg. Press 2ndF SET UP , QUIT to exit the menu. ) b a ▼ 2. To find the solutions of 3sin2 x – 4sin x + 1 = 0, press 3 2 – 4 sin X/θ/T/n + 1 ( sin X/θ/T/n ENTER . 3. Set your viewing window to the principal values for sine by pressing WINDOW (–) 9 0 ENTER 9 0 ENTER 1 0 ENTER (–) 1 ENTER 1 ENTER . Press GRAPH to view the graph. 4. To save this original graph and viewing window, press ZOOM , touch G STO, and double touch 1 StoWin. 5. Set the zoom factors by pressing ZOOM , touching B FACTOR, pressing ENTER 5 ENTER 5 ENTER . Press GRAPH to exit the menu. intercept by continuing to press the 7. ▼ 6. To find the left intercept press TRACE and move the cursor near this key. Now, press ZOOM , touch A ZOOM, and double touch 3 In to zoom in on the left intercept and solution of this equation. Press TRACE and ▼ Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. ENTER 1 to trace near and approximate the intercept. 8. Return to the original screen by pressing ZOOM , touching H RCL, and double touching 1 RclWin. Trace to find the right intercept. 72 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 6.2 SOLVING TRIGONOMETRIC EQUATIONS To solve a trigonometric equation graphically using the CALC feature: 1. Press Y= CL to access and clear Y1. Set the calculator to radian mode by pressing 2ndF Press 2ndF SET UP , touching B DRG, and double touching 2 Rad. QUIT to exit the menu. b a 2 ▼ 2. To find the solutions of 2sin2 x – cos x – 1 = 0, press 2 – cos X/θ/T/n – 1 1 2ndF ENTER 1 π ENTER 2ndF sin X/θ/T/n ) ENTER . (–) 3. Set your viewing window by pressing WINDOW ENTER ( π ÷ 4 2ndF ENTER (–) π ÷ 1 ENTER 2 ENTER . Press GRAPH to view the graph. 4. To "calculate" an intercept (solution), press 2ndF CALC and pressing 2ndF CALC and double touching 5 X_Incpt again. To solve an equation with functions on both the left and right sides: 1. Press Y= CL to access and clear Y1 of an old expression. To find the solutions of 2tan x sin x + 2sin x = tan x + 1, you will enter the left side as Y1, by pressing 2 tan X/θ/T/n sin X/θ/T/n 2. Enter the right side as Y2, by pressing tan + 2 2ndF ENTER 1 π ENTER 2ndF π ÷ sin X/θ/T/n 3. Set your viewing window by pressing WINDOW ENTER 2 4 (–) ENTER + X/θ/T/n 1 ENTER . ENTER . 2ndF π ÷ (–) 2 ENTER 2 ENTER . Press GRAPH to view the graph. 4. There appears to be four solutions to the equation. To "calculate" the intersections (solutions), press TRACE , move the tracer right until it appears on the screen. Next, press 2ndF CALC and double touch 2 Intsct. Do this several times. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 73 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. double touch 5 X_Incpt. You can calculate the other two intercepts by NAME _____________________________________________________ CLASS __________ DATE __________ 6.3 SOLVING TRIGONOMETRIC EQUATIONS Activity 1 1. 2sin2 x + 3sin x – 2 = 0 _______________ degrees 2. 2sin x + √ 3 = 0 _______________ degrees 3. 4cos2 x = 1 _______________ degrees 4. sin 2x = cos x _______________ radians 5. tan x + cot x = 2 _______________ radians Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. Activity 2 74 Solve the following with the zoom feature. Solve the following with the CALC feature. 1. sin x + cos x = 0 _______________ degrees 2. sin2 x = cos x – 1 _______________ degrees 3. 3cos 2x – 5cos x = 1 _______________ radians 4. √ 3cot x sin x + 2cos2 x = 0 _______________ radians 5. 4tan x + sin 2x = 0 _______________ radians Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 7.1 APPLICATIONS WITH RIGHT TRIANGLES Steps for using the calculator to solving applications with right triangles 1. Turn the calculator on and set it to degree mode by pressing 2ndF touching B DRG, and double touching 1 Deg. Press 2ndF SET UP , QUIT to exit the set up menu. 2. Given a right triangle with A = 3 and C = 6, find B and θ using the SOLVER. To access the SOLVER, press 2ndF SOLVER , and press CL to delete any equation that may appear on the screen. ALPHA θ ▼ A 3. Type in the equation sin θ = C with the keystrokes sin ALPHA = a/b ALPHA A ALPHA C . SOLVER , touch A METHOD, and double touch 1 Equation to choose the Equation solver. Press ENTER and see the list of variables. 5. Press ▼ to move the blinking cursor over the A and press 3 Type in the value of C by pressing 6 ▲ ▲ 2nd ENTER . ENTER . Solve for θ by pressing EXE . ▼ ▼ ▼ 6. To use the SOLVER to find B, edit the current equation to the new equation cos θ = B . Do this by pressing CL CL to display the current equation sin A C θ = C . With the cursor over the s in sin, press cos to insert cos. Press DEL to delete sin. Press to move the cursor over the A in the numerator of the fraction. Press ALPHA B to insert B. Press DEL to delete A. 7. Press ENTER to see the variable list. The value for θ and C will remain as before. Press ▼ to position the cursor over the B and press 2ndF EXE to find B. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 75 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 4. Press 2ndF NAME _____________________________________________________ CLASS __________ DATE __________ 7.2 APPLICATIONS WITH RIGHT TRIANGLES A rectangle has length 16 cm and width 8.6 cm. Find the degree measure of angle A. Determine the length of the diagonal of the rectangle. 8.6 A( 16 1. Press + – × ÷ 2ndF SOLVER to access the SOLVER, and press CL to delete the equation that is on the screen. ALPHA θ ALPHA = Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. ▼ A 2. Enter the equation tan θ = B by pressing tan a/b ALPHA A ALPHA B . 3. Press ENTER to view the variable list. 4. Enter 8.6 for A and 16 for B. (The value of θ will be listed as the one that was used in the previously entered equation.) 5. Position the cursor over θ and press 2ndF EXE to find θ. 6. To determine the length of the diagonal of the rectangle, return to the equation by pressing CL CL and edit the current equation to read sin θ = A/C. 7. Press ENTER to see the variable list. The value of θ should be as before, so enter 8.6 for A and solve for C. 76 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 7.3 APPLICATIONS WITH RIGHT TRIANGLES Use the solver with sine, cosine, and tangent functions. 1. Let ABC be a right triangle with AB=32, BC=12.5, and right angle at C. Find the degree measure of the acute angle at A and side AC. B 32 12.5 A C 2. A wheelchair ramp will be built so that it rises to a height of 3 feet above level ground. If the end of the base of the ramp is 12 feet from the building, how long is the ramp? What is the degree measure of the angle at the end of the ramp and level ground? 12 3. A guy wire for a television antenna makes an angle of 60° with the mast. If the wire is attached to the antenna 50 feet from the base of the antenna, use the secant function to determine the length of the wire. 60° 50 ft 4. John is lying on the beach 500 feet from his hotel and looking at Paula who is standing on a balcony. If the balcony is 450 feet above ground level, use the cotangent function to determine the angle between the beach and John’s line of sight. Paula 450 John ) θ 500 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 77 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 3 NAME _____________________________________________________ CLASS __________ DATE __________ 8.1 LAW OF SINES Use the Graphic Solver with the Law of Sines. 1. Set the calculator to degrees by pressing 2ndF and double touching 1 Deg. Press 2ndF sin ALPHA B ALPHA ALPHA A A B CL Enter sin X = sin Y ALPHA X ALPHA = a/b ▼ sin QUIT to exit the set up screen. SOLVER ▼ by pressing a/b ▼ 2. To access the SOLVER, press 2ndF SET UP , touching B DRG, Y . 3. Store it into memory by pressing 2ndF SOLVER , touching C SAVE, pressing ENTER , and typing in LAWSINES. Press ENTER to store the formula. 4. Press 2ndF SOLVER , touch A METHOD, double touch 3 Graphic to Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. choose the Graphic Solver. Press ENTER for the list of variables. 5. With the blinking cursor over the A, type in 95 and press ENTER . Next, type 15 for x, the length of the side opposite the 95°, and press ENTER . Enter 48 for B in the same manner. Solve for y by positioning the blinking cursor over the y and press 2ndF EXE . 6. You must enter the range values for the unknown (from BEGIN to END) within which you expect to find the solution. Let the BEGIN = 5 by pressing 5 ENTER . Enter a value for END that you do not expect y to exceed, say 25, by pressing 2 5 ENTER . Press 2ndF EXE and an autoscaled plot will be drawn showing graphs of the left and right sides of the current equation. 7. The calculator will now display a blinking box in the upper right-hand corner of the viewing screen while it looks for the intersection of the two graphs. The solution is displayed at the bottom of the screen. 78 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 8.2 LAW OF COSINES Use the Newton Solver with the Law of Cosines. 1. Access the SOLVER with 2ndF SOLVER (or press CL twice if you are still in the SOLVER). Press CL to delete the current equation. Return to the Equation Solver by pressing 2ndF SOLVER , touch A METHOD, and double touch 1 Equation. 2. Next enter the Law of Cosines by pressing ALPHA ALPHA Y × X cos 2 x + ALPHA ALPHA Y 2 x – 2 Z ALPHA x2 X ALPHA × = ALPHA θ . 3. Store this equation in the calculator. 4. Given a triangle in which two sides of lengths 13 cm and 8 cm enclose a 50° 5. Press ENTER to view the list of variables. Press ▼ to position the cursor on X and enter 13. Next, enter 8 for Y and 50 for θ. Press ▲ to move the cursor over Z and press 0 6. Press 2ndF ENTER . EXE to solve for Z. Notice that the SOLVER has automatically changed to the Newton Solver. You should supply an initial estimate of the solution in the START position (0 is the default value) and a step size STEP (.001 is the default value). 7. Press 2ndF EXE to find the solution. (If you receive an error message, try changing the START value or increasing the STEP value.) 8. Notice the left and right sides of the equation are displayed as equal values, and the L – R = 0. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 79 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. angle. Find the length of the third side. NAME _____________________________________________________ CLASS __________ DATE __________ 8.3 LAW OF SINES AND LAW OF COSINES Use the SOLVER with the Law of Sines. ❂ 1. A 50 foot tree has been damaged ) in a storm and is inclined at an B° angle of θ° (from the vertical) directly away from the sun. Find 50 ft if the tree casts a shadow 40 feet θ° long on level ground when the 60° angle of elevation of the sun is 60°. 40 ft Use the SOLVER with the Law of Cosines. B 2. A bridge is to be built between Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. points B and C across a lake. A surveyor using a transit determines 82° A the measure of the angle at point A d to be 82°. The distance between points A and B and between points A and C is known to be 125 yards. 125 Find the distance across the lake that the bridge will span. C 3. Two adjacent sides of a parallelogram are of length 8 and 12. The length of 8 the shorter diagonal of the parallelogram is 10.5. Find the angle opposite the shorter diagonal. ) 10.5 θ 12 80 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 9.1 AREA OF A TRIANGLE 1. Set the calculator to operate in degrees with 2ndF and double touch 1 Deg. Press 2ndF 2. Access the SOLVER by pressing 2ndF SET UP , touch B DRG, QUIT to exit the set up screen. SOLVER , and press CL to delete any equation on the screen. a/b 1 4. Press 2ndF ▼ = ▼ 3. Type in the equation A = 2 1 B•H with the keystrokes: ALPHA 2 ALPHA B × ALPHA H A ALPHA SOLVER , touch A METHOD, and double touch 1 Equation to choose the Equation solver. Press ENTER to see the list of variables for your equation. 5. Find the area of a triangular garden plot with base = 4 meters and height = Type in 8.35 for H and press ENTER . Press ▲ twice to move the prompt to the A variable. Press 2ndF EXE to find the area, 16.7 square meters. 6. Save the formula by pressing 2ndF SOLVER and touching C SAVE. Enter a name and press ENTER . = ALPHA θ . a/b 1 2 ▼ ALPHA ▼ 7. Enter A = 1 B•C•sin θ by pressing CL a couple times to return to and delete 2 the current equation. Type in the equation with the keystrokes: ALPHA A ALPHA B × ALPHA C × sin 8. Consider the triangle where sides of lengths 4.8 and 5.73 enclose a 47° angle. Press ENTER to access the variable list, enter 5.73 for B, 4.8 for C (or vice-versa), and 47 for θ. Move the cursor to the A variable. Press 2ndF EXE to solve for the area. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 81 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 8.35 meters, use ▼ to move the blinking cursor over B and type in 4. NAME _____________________________________________________ CLASS __________ DATE __________ 9.2 AREA OF A TRIANGLE 1. Continue with the formula from Blackline Master 9.1. Press 2ndF SOLVER , touch A METHOD, and double touch 3 Graphic to choose the Graphic solver. 2. To start the problem from the beginning, set A to 0. Move the cursor to the A variable and press 2ndF EXE to see the Graphic solver variable range screen. 3. Since area is always nonnegative, 0 could be used for BEGIN. Enter END a value that you do not expect area to exceed, say 25. (A little bit of estimation helps here! If your value for END is too small, increase it and try again.) 4. Press 2ndF EXE and an autoscaled graph of the two sides of the equation Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. is drawn and the solution is obtained. 5. Enter Heron’s formula, A = √ [S(S-X)(S-Y)(S-Z)], press CL , and type the following: ALPHA A ALPHA = 2ndF √ ALPHA S ( ALPHA S – ALPHA X ) S – ALPHA Z ) . ( ALPHA S – ALPHA Y ) ( ALPHA 6. Note that the square root symbol extends over the entire right hand side of the equation, automatically grouping the expression under it. 1 7. You could enter the semiperimeter, S = 2 (X + Y + Z), in a separate formula in the SOLVER, but it is probably easier to find the value of S in the computation mode and then use Heron’s formula in the SOLVER. Press ENTER to view the variable list. 8. Use Heron's formula to solve some area problems. 82 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 9.3 AREA OF A TRIANGLE Use the solver to find the area of triangle. 1. Two legs of a right triangle (neither of which is the hypotenuse) are of length 25 and 38. Find the area of the triangle. 2. Triangle ABC has two sides of lengths 13 √ 3 cm and 18 cm that enclose angle q. The area of triangle ABC is 132.95 square cm. Find the height h of triangle ABC. Find the measure of angle θ. B 18 h )θ 13 √ 3 C 3. A wooded lot has the shape of ∆STR in the figure below. The highway department plans to purchase the region shown as ∆MOR for frontage on a proposed interstate highway. A survey of the region shows the distance from point M to point O is 60 feet, the distance from R to M is 85 feet, and the angle enclosed by these two lengths is 65°. The deed for the lot indicates that the distance from S to T is 170 feet and the distance from S to R is 245 feet. The owner of the lot has determined that the area of the four-sided region remaining after the highway department purchase is 16,244 square feet. Find the area of the region the highway department plans to purchase. Find the measure of angle at the back corner of the wooded lot. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 83 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. A NAME _____________________________________________________ CLASS __________ DATE __________ 9.4 AREA OF A TRIANGLE Use the solver with Heron’s formula. 4. A spider web in the shape of a regular octagon is such that each of the eight triangular regions shown in the figure has the same shape and size as the shaded region. If the distance from the center of the web to point R is 8 inches and equals the distance from the center of the web to point S, find Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. the total area enclosed by the web. Find the perimeter of the web. 5. Yum-Yum Burgers Corporation is considering purchasing a corner lot in town for a drive-through hamburger restaurant. They have decided to by the land if the price is $9000 or less. If the lot is triangular in shape with sides of lengths 100 feet, 50 feet, 72 feet and the owner is asking $5 per square foot, will Yum-Yum Burgers purchase the lot? 6. Jim is lost in the forest. The police have determined that they should search the shaded region in the figure below. If each unit on the grid represents one mile, what is the area of region that the police will search? 84 Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 10.1 AREA OF A CIRCULAR SECTOR A lawn sprinkler is set so that the region of the lawn that is watered is approximately a sector of a circle of radius 6 feet. If the central angle of the π sector is radians, find the area of the region that is watered by the sprinkler. 5 1. Set the calculator to radian mode by pressing 2ndF B DRG, and double touching 2 Rad. Press 2ndF SET UP , touching QUIT to exit the set up screen. 2. Access the SOLVER by pressing + – × ÷ 2ndF SOLVER . Press CL to delete any equation that appears on the screen. A ALPHA = 1 ▼ 1 3. Type in the equation A= 2 R2θ by pressing ALPHA a/b 2 ALPHA R x2 ALPHA θ . SOLVER , touch A METHOD, and double touch 1 Equation. Press ENTER and see the list of variables. 5. Use ▼ to place the cursor over R, type in 6, and press ENTER . Type in θ by pressing 2ndF π ÷ 5 on the A variable. Press 2ndF ENTER . Press ▲ twice to place the cursor EXE to find the area in square feet. 6. Use the Newton solver by pressing 2ndF SOLVER , touch A METHOD, and double touch 2 Newton. Reset A to 0 to start the problem from the beginning. The values of R and O are correct for this problem, so use ▲ to place the cursor over the A variable and press 2ndF EXE to view the Newton solver screen. 7. Since area is never negative, you could set START = 0. Leave the STEP size as .001 and remember that you should change it to a slightly larger value if an error message results. Press 2ndF EXE to find the area in square feet. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 85 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. 4. Use the Equation solver by pressing 2ndF NAME _____________________________________________________ CLASS __________ DATE __________ 10.2 AREA OF A CIRCULAR SECTOR 1. Using the problem from Blackline Master 10.1, find the area using the Graphic Solver. 2. Press 2ndF SOLVER , touch A METHOD, and double touch 3 Graphic. 3. Reset A to 0 to start the problem from the beginning. 4. Place the cursor on the A variable and press 2ndF EXE . 5. Recall that BEGIN and END are the lower and upper values between which you feel the solution will lie. Enter a 20 for END so that the intersection of Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. the two graphs is visible on the screen. 6. Press 2ndF 86 EXE to find the area. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 NAME _____________________________________________________ CLASS __________ DATE __________ 10.3 AREA OF A CIRCULAR SECTOR Use the Equation or Newton solver to find the area of a circular sector. 1. Find the area of a circular sector subtended by a central angle of 0.6π radians in a circle of radius 5 yards. 2. A sector of a circle has an area equal to 36 square inches. If the arc of the sector subtends an angle of 55° at the center of the circle, find the radius of the circle. 3. A large pizza in the shape of a circle has a diameter of 20 inches. The pizza is cut into 8 slices of equal area. What is the area of each piece of pizza? 4. A circular clock has the tip of the hour hand touching the rim of the clock. What is the area of the sector formed when the hour hand moves from 1 o’clock to 5 o’clock? 5. A cherry cheesecake is in the shape of a circle. If the diameter of the pan holding the cheesecake is 9 inches, and the cheesecake is cut into 10 equal pieces, find the area covered by the cherry topping on one piece of the cheesecake. Blackline Masters/TRIGONOMETRY USING THE SHARP EL-9600 87 Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. Use the Graphic solver to find the area of a circular sector. KEYPAD FOR THE SHARP EL-9600 GRAPHING CALCULATOR STAT PLOT SPLIT TBL SET SUB FORMAT CALC Y= GRAPH TABLE WINDOW ZOOM TRACE SOLVER CLIP SHIFT/CHANGE + – × ÷ SLIDE SHOW EZ OFF PRGM DRAW ON MATRIX STAT A-LOCK TOOL = INS SET UP ALPHA MATH DEL BS -1 sin A -1 cos B tan C cos sin , a b L v Q L4 V L1 2 1 < O 0 i SPACE L3 ▼ log F In π K OPTION VARS X/θ/T/n { O S LIST X e } P ) T FINANCE U Y EXE + 3 ENTRY CL ex ( 6 W L2 N L6 5 4 2 9 R L5 x QUIT D 10x E STO w 8 -1 RCL J I ab M 7 x tan H a G u ▼ + -1 Exp ▼ 2nd F 88 Equation Editor ▼ Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only. EL-9600 : (–) Keypad/TRIGONOMETRY USING THE SHARP EL-9600 ANS ENTER Z SOLUTIONS TO SELECTED ACTIVITIES 2. EVALUATING TRIGONOMETRIC FUNCTIONS ADDITIONAL PROBLEMS 1. .8910 2. .7813 3. 2 4. .9563 5. .4816 6. -1.4142 BLACKLINE MASTERS 2.3-2.4 ACTIVITY 1 ACTIVITY 2 1. .9848 7. -.5878 2. .9848 8. -.8660 3. 2.7475 9. 1 4. 1.2208 10. -1.4142 5. 1.0785 11. -2 6. 1.1918 12. -2.0765 ACTIVITY 3 ACTIVITY 4 13. 3090 19. .5878 14. .8387 20. -.9010 15. 1.0355 21. -1 16. 1.0038 22. 2.9238 17. -1.2208 23. 2.6131 18. .7002 24. .1438 3. GRAPHICAL VERIFICATION OF FORMULAS OR IDENTITIES ADDITIONAL PROBLEMS 1. Identity 2. Identity 3. Identity BLACKLINE MASTER 3.3 Identity BLACKLINE MASTER 3.4 Identity BLACKLINE MASTER 3.5 ACTIVITY 3 ACTIVITY 4 Identity 1. Identity 2. Identity 3. Identity 4. AMPLITUDE, PERIOD AND PHASE SHIFT ADDITIONAL PROBLEMS 1. flipped over, rising faster 2. period is 3π π 3. shifted right 2 π 4. shifted left and up 2 2 BLACKLINE MASTER 4.1 3 sin x has an amplitude of 3, -2sin x is flipped over with an amplitude of 2 BLACKLINE MASTER 4.2 π The function has moved 3 2 to the right, and down 2 BLACKLINE MASTER 4.3 1. larger amplitude 2. smaller amplitude 3. smaller period 4. larger period π 5. moved right 6 π 6. moved left 6 π 7. larger amplitude, moved left 12 , smaller period 8. pinched in, larger period 2π 9. larger period, moved right 3 10.moved farther from x-axis, smaller period 11.moved closer to x-axis, moved right π 6 5. INVERSE TRIGONOMETRIC FUNCTIONS ADDITIONAL PROBLEMS 1. -44.427° 2. 60° 3. -63.4349° 4. -30° 5. .2014 6. 2.2143 7. 1.2490 8. .3218 9. .8411 BLACKLINE MASTER 5.3 1. 53.1301° 2. 126.8698° 3. 63.4349° 4. 2.4484 5. .6289 6. -.7854 6. SOLVING TRIGONOMETRIC EQUATIONS ADDITIONAL PROBLEMS 1. -60° and 60° 2. 90° 3. -90, 30°, 150° 4. 3.1416 Answer Key/TRIGONOMETRY USING THE SHARP EL-9600 89 5. 0, 3.1416 6. .5236, 2.618 BLACKLINE MASTER 6.1 7. 19.4° 8. 90° BLACKLINE MASTER 6.2 ACTIVITY 1 1.0472, 3.1416, -1.0472 ACTIVITY 2 2.3562, 2.618, -.7854, .5236 BLACKLINE MASTER 6.3 ACTIVITY 1 1. 30°, 150° 2. -120°, -60° 3. 60°, 120° 4. .5236, 1.5708, 2.6180 5. .7854 ACTIVITY 2 1. -45°, 135° 2. 0° 3. -2.0944, 2.0944, 4. 1.5708, 2.6180, 3.6652, 4.7124 5. 0 7 APPLICATIONS WITH RIGHT TRIANGLES ADDITIONAL PROBLEMS 1. 5.945 2. 37.859 3. 36.87° 4. 3.849 5. No, one possible explanation is that sin > 1 which is not possible 6. 216.306 8 LAW OF SINES AND LAW OF COSINES ADDITIONAL PROBLEMS 1. 1447.761 2. 766.986 3. 502.341 4. 1521.317 BLACKLINE MASTER 8.1 7. 11.1898 BLACKLINE MASTER 8.2 7. 9.9649 BLACKLINE MASTER 8.3 1. 13.854° 2. 200 yards 3. 59.395° 9 AREA OF A TRIANGLE ADDITIONAL PROBLEMS 1. 30.311 2. 1.5 3. 190.84 4. 12 5. no intersection, sin > 1 impossible BLACKLINE MASTER 9.1 8. 10.06 BLACKLINE MASTER 9.3 1. 475 2. 11.809, 41° 3. 2311.085, 63° BLACKLINE MASTER 9.4 4. 181 5. 8521.68 < $9000 6. 48.5 BLACKLINE MASTER 7.1 5. 30° 7. 5.196 10 AREA OF A CIRCULAR SECTOR ADDITIONAL PROBLEMS 1. 42.412 2. 6.981 3. 2.611 radians or 149.606° BLACKLINE MASTER 7.2 5. 28.258° 7. 18.165 BLACKLINE MASTER 10.1 5. 11.3097 7. 11.3097 BLACKLINE MASTER 7.3 1. 22.993°, 29.458 2. 12.369, 14.036° 3. 100 feet 4. 41.987° BLACKLINE MASTER 10.3 1. 23.562 2. 8.661 3. 39.27 4. π r2/3 5. 6.362 90 Answer Key/TRIGONOMETRY USING THE SHARP EL-9600 TEACHING NOTES Teaching Notes/TRIGONOMETRY USING THE SHARP EL-9600 91 TEACHING NOTES 92 Teaching Notes/TRIGONOMETRY USING THE SHARP EL-9600 SHARP TRIG COVER 02.2.19 10:55 AM Page 2

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