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NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 12-1 Study Guide and Intervention Trigonometric Functions in Right Triangles Trigonometric Functions for Acute Angles Trigonometry is the study of relationships among the angles and sides of a right triangle. A trigonometric function has a rule given by a trigonometric ratio, which is a ratio that compares the side lengths of a right triangle. Trigonometric Functions in Right Triangles If θ is the measure of an acute angle of a right triangle, opp is the measure of the leg opposite θ, adj is the measure of the leg adjacent to θ, and hyp is the measure of the hypotenuse, then the following are true. sin θ = csc θ = opp cos θ = hyp hyp sec θ = opp adj hyp hyp adj tan θ = opp cot θ = adj adj opp 𝟑 Example: In a right triangle, ∠ B is acute and cos B = 𝟕. Find the value of tan B. Step 1 Draw a right triangle and label one acute angle B. Label the adjacent side 3 and the hypotenuse 7. Step 2 Use the Pythagorean Theorem to find b. 𝑎2 + 𝑏 2 = 𝑐 2 Pythagorean Theorem 32 + 𝑏 2 = 72 a = 3 and c = 7 9 + 𝑏 2 = 49 Simplify. 2 Subtract 9 from each side. b = √40 or 2√10 Take the positive square root of each side. 𝑏 = 40 Step 3 Find tan B. tan B = opp adj Tangent function tan B = 2√10 3 Replace opp with 2√10 and adj with 3. Exercises Find the values of the six trigonometric functions for angle θ. 1. 2. 3. In a right triangle, ∠ A and ∠ B are acute. 7 4. If tan A = 12 , what is cos A? Chapter 12 1 5. If cos A = 2 , what is tan A? 5 3 6. If sin B = 8 , what is tan B? Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 12-1 Study Guide and Intervention (continued) Trigonometric Functions in Right Triangles Use Trigonometric Functions You can use trigonometric functions to find missing side lengths and missing angle measures of right triangles. You can find the measure of the missing angle by using the inverse of sine, cosine, or tangent. Example: Find the measure of ∠ C. Round to the nearest tenth if necessary. You know the measure of the side opposite ∠ C and the measure of the hypotenuse. Use the sine function. opp sin C = hyp 8 sin C = 10 sin−1 8 10 = m∠ C 53.1° ≈ m∠ C Sine function Replace opp with 8 and hyp with 10. Inverse sine Use a calculator. Exercises Use a trigonometric function to find each value of x. Round to the nearest tenth if necessary. 1. 2. 3. 4. 5. 6. Find x. Round to the nearest tenth if necessary. 7. 8. Chapter 12 9. 6 Glencoe Algebra 2