22-2 and 22-3: Trig Ratios Objectives: 1. To discover and use the three main trigonometric ratios • • • • Assignment: P. 311: 12-14 P. 314: 7-9 P. 317-318: 5-10 More on Tangent Ratios Worksheet Objective 1 You will discover and use the three main trigonometric ratios Warm-Up 1 Find the value of x. Warm-Up 1 Find the value of x. History Lesson Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles. History Lesson Early mathematicians discovered trig by measuring the ratios of the sides of different right triangles. They noticed that when the ratio of the shorter leg to the longer leg was close to a specific number, then the angle opposite the shorter leg was close to a specific number. Example 1 In every right triangle in which the ratio of the shorter leg to the longer leg is 3/5, the angle opposite the shorter leg measures close to 31. What is a good approximation for x? Example 2 In every right triangle in which the ratio of the shorter leg to the longer leg is 9/10, the angle opposite the shorter leg measures close to 42. What is a good approximation for y? Trig Ratios The previous examples worked because the triangles were similar since the angles were congruent. This means that the ratios of the sides are equal. In those cases we were using the tangent ratio. Here’s a list of the three you’ll have to know. sine cosine tangent Investigation 1 Use the Geogebra Activity to discover the three main Trigonometric ratios sine, cosine, and tangent. Summary A side adjacent Θ B side opposite Θ C sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent Summary A side adjacent Θ B side opposite Θ C sin Oh Hell cos Another Hour tan Of Algebra SohCahToa Soh sin opposite hypotenuse Cah cos adjacent hypotenuse Toa tan opposite adjacent Example 3 Find the values of the six trig ratios for α and β. Activity: Trig Table On the previous example, we knew all the sides of the triangle, and we just listed the three trig ratios for those sides using a generic angle. Usually, though, you know the angle, and you want to find a side. Nowadays, we would use a calculator to find the sine or tangent of an angle. In the long, dark years before the calculator, people had to find their trig ratios in a table. Activity: Trig Table In the 1500s, Georg Rheticus, a student of Copernicus, was the first to define the six trig functions in terms of right triangles. He was also the first to start a book of values for these ratios, accurate to ten decimal places to be used in astronomical calculations. Activity: Trig Table Of course, he died before it was completed, and it was up to his student, Valentin Otto, to finish the 1500 page book. We’re going to do something similar, but ours will only be accurate for 3 decimal places, and probably wouldn’t be too reliable for astronomical calculations. Investigation 2 Use the triangle below to calculate the sine, cosine, and tangent of 20. C m AC = 15 cm m CB = 5 cm 20 A m AB = 14 cm B Investigation 2 Use the triangle below to calculate the sine, cosine, and tangent of 20. C m AC = 14.9 cm m CB = 5.1 cm 20 A m AB = 14.0 cm B Investigation 2 Use the triangle below to calculate the sine, cosine, and tangent of 20. C m AC = 14.92 cm m CB = 5.10 cm 20 A m AB = 14.02 cm B Investigation 2 Use the triangle below to calculate the sine, cosine, and tangent of 20. C m AC = 14.923 cm m CB = 5.104 cm 20 A m AB = 14.023 cm B Activity: Trig Table Step 1: On a sheet of graph paper or scratch paper, use your protractor to make as large a right Δ𝐴𝐵𝐶 as possible with 𝑚∠𝐵 = 90°, 𝑚∠𝐴 = 20°, and 𝑚∠𝐶 = 70°. Activity: Trig Table Step 2: Measure sides 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶 with your ruler to the nearest millimeter. Activity: Trig Table Step 3: Set up a table of values like so: θ 20° 70° sin θ cos θ tan θ Activity: Trig Table Step 4: Now use your calculator to round each calculation to the nearest thousandths place. θ 20° 70° sin θ cos θ tan θ Activity: Trig Table Step 5: Finally, let’s check your values with those from the calculator. For sin, cos, and tan 1. Make sure your calculator is set to DEGREE in the MODE menu. 2. Use one of the 3 trig keys. Get in the habit of closing the parenthesis. Example 4 To the nearest meter, find the height of a right triangle if one acute angle measures 35° and the adjacent side measures 24 m. Example 5 To the nearest foot, find the length of the hypotenuse of a right triangle if one of the acute angles measures 20° and the opposite side measures 410 feet. Example 6 Use a special right triangle to find the exact values of sin(45°) and cos(45°). Example 7 Find the value of x to the nearest tenth. 1. x = 2. x = 3. x = 22-2 and 22-3: Trig Ratios Objectives: 1. To discover and use the three main trigonometric ratios • • • • Assignment: P. 311: 12-14 P. 314: 7-9 P. 317-318: 5-10 More on Tangent Ratios Worksheet

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