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• How do I use the sine, cosine, and tangent ratios to solve triangles? 5.2 Apply the Tangent Ratio Trigonometric Ratios B c a C b opposite a sin A = hypotenuse c A b adjacent = cos A = hypotenuse c opposite a tan A = = adjacent b 5.2 Apply the Tangent Ratio Tangent Ratio Let rABC be a right triangle with acute A. The tangent of A (written as tan A) is defined as follows: B C A BC length of leg opposite A tan A AC length of leg adjacent t o A _______ 5.2 Apply the Tangent Ratio Example 1 Find tangent ratios Find tan X and tan Y. Write each answer as a fraction and as a decimal rounded to four places. Y 8 Z 17 15 0.5333 ______ X YZ 8 opp. X tan X XZ _______ 15 adj. to X _______ XZ 15 opp. Y 1.875 tan Y ______ YZ _______ 8 adj. to Y _______ In the right triangle, rXYZ, X and Y are _______________ complementary angles. You can see that the tangent ratios of the _________________ angles are complementary ___________. reciprocals 5.2 Apply the Tangent Ratio Example 2 Find a leg length Find the value of x. Use the tangent of an acute angle to find a leg length. opp. o tan 25 __________ adj. x o tan 25 _______ 12 o ___ tan 25 x 12 ___ 0.4663 x 12 ______ _____ 5.6 x x 25o 12 Write a ratio for tangent of 25o. Substitute. Multiply each side by ____. 12 Use a calculator to find tan 25o. Simplify. 5.2 Apply the Tangent Ratio Checkpoint. Complete the following exercises. 1. Find tan A and tan B. Round your answer to four decimal places. A tan A 18 24 0.75 tan B 24 1.3333 18 C 24 18 30 B 5.2 Apply the Tangent Ratio Checkpoint. Complete the following exercises. 2. Find the value of x. Round to the nearest tenth. x 38 20 x tan 38 20 o 20 20 x 20 0.7813 x 15.6 o 5.2 Apply the Tangent Ratio Example 3 Compare the tangent ratios for similar triangles Find tan X and tan Y for similar triangles. Then compare the Y tangent ratios. Y b Z c a 3b X b tan X a _______ a tan Y b ______ 3c X 3a tan X 3b b a _______ ______ 3a tan Y 3a a ______ 3b ______ b Z equivalent The values of tan X and tan Y for the similar triangles are ________. 5.2 Apply the Tangent Ratio Example 4 Estimate height using tangent Find the height h of the flagpole to the nearest foot. opp. tan 70 adj. _______ o tan 70o h 7 _______ ___ 7 tan 70o ___ h ___ 19 h h Write ratio for tangent of 70o. Substitute. 70 o 7 ft Multiply each side by ___. 7 Use a calculator to simplify. 19 feet tall. The flagpole is about _____ 5.2 Apply the Tangent Ratio Checkpoint. Complete the following exercises. 3. Find the height h of the flagpole in Example 4 to the nearest foot if the angle is 75o. 7 h tan 75 7 o 7 h 7 3.7321 h 26 ft h 75o 7 ft 5.2 Apply the Tangent Ratio Pg. 173, 5.2 #1-25 5.3 Apply the Sine and Cosine Ratios Pg. 218, 6.4 #1-32