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15 More Trigonometric Identities Worksheet Concepts: • The Addition and Subtraction Identities for Sine and Cosine • The Cofunction Identities for Sine and Cosine • Identities That You Should Learn (Section 7.3) 1. Evaluate the six trigonometric functions at θ = calculator. π . 12 Do not use your 2. Simplify cos(π + x). 3. Prove the identity tan π 2 − x = cot(x). 4. (Exercise 47 in Section 7.2 of your textbook) The terminal ray of x is in the first quadrant, the terminal ray of y is 12 in the second quadrant, sin(x) = 45 , and cos(y) = − 13 . (a) Find the exact values of sin(x + y), cos(x + y), and tan(x + y). (b) In which quadrant is the terminal ray of the angle x + y found? (Assume all angles are in standard position.) π π . 5. Find the exact value of sin −π+ 4 6 6. Express cos(α + β + γ) in terms of sines and cosines of α, β, and γ. 7. Prove that the difference quotient for g(x) = cos(x) is equal to cos(h) − 1 sin(h) cos(x) − sin(x) . h h 8. Express sin2 (3x) in terms of constants and first powers of the cosine function. 1 9. Express cos4 (5x) in terms of constants and first powers of the cosine function. 2 x 10. Express sin in terms of constants and first powers of the cosine 2 function. 2 11. (Exercise 9 from section 7.3 of your textbook) A rectangle is inscribed in a semicircle of radius 3 inches and the radius to the corner makes an angle of t radians with the horizontal, as shown in the picture. • (x, y) t (a) Express the horizontal length, vertical height, and area of the rectangle in terms of x and y. (b) Express x and y in terms of sine and cosine (NOTE: This is not a unit circle.) (c) Use parts (a) and (b) and suitable identities to show that the area of the rectangle is given by A = 9 sin(2t). 12. (Exercise 90 from section 7.3 of your textbook) To avoid a steep hill, a road is being built in straight segments from P to Q and from Q to R; it makes a turn of t radians at Q as shown in the figure. The distance from P to S is 40 miles, and the distance from R to S is 10 miles. Use suitable trigonometric functions to express: (a) c in terms of b and t. [Hint: Place the figure on a coordinate plane with P and Q on the x-axis, with Q at the origin. Then what are the coordinates of R?] (b) b in terms of t (c) a in terms of t [Hint: a = 40 − c; use parts (a) and (b).] (d) Use parts (b) and (c) and a suitable identity to show that the length a + b of the road is t 40 + 10 tan . 2 3 R STEEP HILL b 10 t P a Q 40 4 c S