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13.1 Chapter 13: Trigonometric Ratios and Functions Section 13.1 1 Key Concept Section 13.1 2 Key Concept Section 13.1 3 EXAMPLE 1 Evaluate trigonometric functions Evaluate the six trigonometric functions of the angle θ. EXAMPLE 1 Evaluate trigonometric functions Evaluate the six trigonometric functions of the angle θ. SOLUTION From the Pythagorean theorem, the length of the hypotenuse is √ 52 + 122 = √ 169 = 13. sin θ = opp 12 = hyp 13 csc θ = hyp 13 = 12 opp EXAMPLE 1 Evaluate trigonometric functions cos θ = adj = hyp 5 13 tan θ = opp = adj 12 5 sec θ = 13 hyp = 5 adj cot θ = 5 adj = 12 opp EXAMPLE 2 Standardized Test Practice EXAMPLE 2 Standardized Test Practice SOLUTION STEP 1 Draw: a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. By the Pythagorean theorem, the length x of the other leg is x = √ 72 – 42 = √ 33. EXAMPLE 2 Standardized Test Practice STEP 2 Find the value of tan θ. tan θ = 4 opp = adj √ 33 4 √ 33 = 33 ANSWER The correct answer is B. GUIDED PRACTICE for Examples 1 and 2 Evaluate the six trigonometric functions of the angle θ. 1. ANSWER sin θ = opp = hyp cos θ = adj 4 = 5 hyp tan θ = opp = adj 3 5 3 4 csc θ = hyp 5 = opp 3 sec θ = hyp 5 = 4 adj cot θ = adj 4 = opp 3 GUIDED PRACTICE for Examples 1 and 2 Evaluate the six trigonometric functions of the angle θ. 2. ANSWER sin θ = opp 15 = hyp 17 cos θ = adj 8 = 17 hyp tan θ = opp 15 = adj 8 csc θ = hyp 17 = opp 15 sec θ = hyp 17 = 8 adj cot θ = adj 8 = opp 15 GUIDED PRACTICE for Examples 1 and 2 Evaluate the six trigonometric functions of the angle θ. 3. ANSWER sin θ = opp 5 = hyp 5√2 cos θ = adj 5 = hyp 5√2 tan θ = opp = adj 5 =1 5 csc θ = hyp 5√2 = opp 5 sec θ = hyp 5√2 = adj 5 cot θ = adj 5 =1 = opp 5 GUIDED PRACTICE 4. for Examples 1 and 2 In a right triangle, θ is an acute angle and cos θ = 7 . What is sin θ? 10 ANSWER sin θ = √ 51 10 Key Concept Section 13.1 14 EXAMPLE 3 Find an unknown side length of a right triangle Find the value of x for the right triangle shown. EXAMPLE 3 Find an unknown side length of a right triangle Find the value of x for the right triangle shown. SOLUTION Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x. adj Write trigonometric equation. cos 30º = hyp √3 2 = x 8 Substitute. EXAMPLE 3 4√3 = x Find an unknown side length of a right triangle Multiply each side by 8. ANSWER The length of the side is x = 4 √ 3 6.93. EXAMPLE 4 Solve ABC. Use a calculator to solve a right triangle EXAMPLE 4 Solve Use a calculator to solve a right triangle ABC. SOLUTION A and B are complementary angles, so B = 90º – 28º = 62º. opp tan 28° = adj a tan 28º = 15 sec 28º = hyp adj c sec 28º = 15 Write trigonometric equation. Substitute. EXAMPLE 4 15(tan 28º) = a 7.98 a Use a calculator to solve a right triangle 15 ( 1 cos 28º 17.0 )=c c ANSWER So, B = 62º, a 7.98, and c 17.0. Solve for the variable. Use a calculator. GUIDED PRACTICE for Examples 3 and 4 Solve ABC using the diagram at the right and the given measurements. 5. B = 45°, c = 5 ANSWER So, A = 45º, b 3.54, and a 3.54. GUIDED PRACTICE 6. for Examples 3 and 4 A = 32°, b = 10 ANSWER So, B = 58º, a 6.25, and c 11.8. GUIDED PRACTICE 7. for Examples 3 and 4 A = 71°, c = 20 ANSWER So, B = 19º, b 6.51, and a 18.9. GUIDED PRACTICE 8. for Examples 3 and 4 B = 60°, a = 7 ANSWER So, A = 30º, c = 14, and b 12.1. HOMEWORK Sec 13-1 (pg 856) 3-28 all Section 13.1 25