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Warm-Up Think about what we’ve learned so far in this chapter. How could you find the value of x in the triangle below? 8.4 Trigonometry Trigonometry • Greek for “triangle measure” • Uses the angles and sides of a triangle • Trigonometric ratio is the ratio of 2 side lengths in a triangle in reference to a specific angle in the triangle *The labels for the sides depend on the angle used *Always use one of the 2 acute angles of the triangle Hypotenuse Hypotenuse: across from the right angle always (side c) Side Adjacent Side Opposite Side c Adjacent Side: “next to” The side adjacent to Angle B is side a The side adjacent to Angle A is side b A Opposite Side: “across from” The side opposite of Angle B is side b The side opposite of Angle A is side a B a b C Sine Ratio 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 sin 𝐴 = 𝑎 𝑐 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑏 𝑐 𝑏 cos 𝐴 = 𝑐 sin 𝐵 = Cosine Ratio Trigonometric Ratios 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎 𝑐 𝑎 tan 𝐴 = 𝑏 𝑏 tan 𝐵 = 𝑎 cos 𝐵 = Tangent Ratio 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 SOH-CAH-TOA 𝒐𝑝𝑝 𝒔𝑖𝑛 = 𝒉𝑦𝑝 𝒂𝑑𝑗 𝒄𝑜𝑠 = 𝒉𝑦𝑝 𝒐𝑝𝑝 𝒕𝑎𝑛 = 𝒂𝑑𝑗 Example 1: Given the triangle, label the sides (opposite, adjacent, hypotenuse) according to the specified angle. a. Angle S Example 1: Given the triangle, label the sides (opposite, adjacent, hypotenuse) according to the specified angle. b. Angle P Example 2: Given the triangle, label the sides (opposite, adjacent, hypotenuse) according to the specified angle. a. Angle R Example 2: Given the triangle, label the sides (opposite, adjacent, hypotenuse) according to the specified angle. b. Angle S Example 1 a) Express sin L as a fraction and as a decimal to the nearest hundredth. Find the 3 trigonometric ratios for angle L. a) sin L b) cos L c) tan L Example 1 d) Express sin N as a fraction and as a decimal to the nearest hundredth. Find 3 trigonometric ratios for angle N. a) sin N b) cos N c) tan N Example 2: a) Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. b) Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundreth Example 3: SOLVE. Round to the nearest tenth. a. x sin 23 15 b. w cos 74 5 Example 3: Round to the nearest tenth. c. d. 3 18 sin 80 tan 60 x y Example 3: Round to the nearest tenth. e. 10 cos12 y f. z tan 68 43 Example 4: Use trigonometric ratios to find the value of the variables. Round to the nearest hundredth. Example 4: Use trigonometric ratios to find the value of the variables. Round to the nearest hundredth. Change location of 5 and x! x 27 5 Exit Slip Find the value of y. Example 5: A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Example 6: The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch? Use trigonometric rations to find the value of the variables. Round to the nearest hundredth Use trigonometric rations to find the value of the variables. Round to the nearest hundredth If sine A = x, then the inverse sine of x is the measure of angle A. Inverse Sine Inverse Trigonometric Ratios *used to find the measure of an angle* 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑛−1 (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) If cos A = x, then the inverse cosine of x is the measure of angle A. Inverse Cosine 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑐𝑜𝑠 −1 (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) Inverse Tangent If tan A = x, then the inverse tangent of x is the measure of angle A. 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑎𝑛−1 (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡) What is the difference between Trigonometric functions and Inverse Trigonometric Functions? • Trigonometric functions (sin, cos, tan) are used to find side lengths • Inverse trigonometric functions (sin-1, cos-1, tan-1) are used to find angle measures Example 5: a) Use a calculator to find the measure of P to the nearest tenth. Example 5: b) Use a calculator to find the measure of D to the nearest tenth. Example 6: Solve each right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Example 6: Solve each right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. *Change 22°