711 7.4 Double-Angle and Half-Angle Identities 1. If sin u = - 7 25 and u is in quadrant IV, find the remaining five trigonometric function values of u. 2. Express cot2 x + csc2 x in terms of sin x and cos x, and simplify. 3. Find the exact value of sin A - 7p 12 B . 4. Express cos1180° - u2 as a function of u alone. 5. If cos A = 3 5 , sin B = - 5 13 , 0 6A6 p 2 , and p6B6 3p 2 , find each of the following. (a) cos1A + B2 (b) sin1A + B2 (c) the quadrant of A + B 6. Express tan A3p 4 + xB as a function of x alone. Verify that each equation is an identity. 7. 1 + sin u cot2 u = sin u csc u - 1 8. sin a p 3 + ub - sin a p 3 - ub = sin u 9. sin2 u - cos2 u sin4 u - cos4 u = 1 10. cos1x + y2 + cos1x - y2 sin1x - y2 + sin1x + y2 = cot x Chapter 7 Quiz (Sections 7.1 – 7.3) 7.4 Double-Angle and Half-Angle Identities ■ Double-Angle Identities ■ An Application ■ Product-to-Sum and Sum-to-Product Identities ■ Half-Angle Identities ■ Verifying an Identity Double-Angle Identities When A = B in the identities for the sum of two angles, the double-angle identities result. To derive an expression for cos 2A, we let B = A in the identity cos1A + B2 = cos A cos B - sin A sin B. cos 2 A = cos1A + A2 2A = A + A = cos A cos A - sin A sin A Cosine sum identity cos 2 A =cos2 A −sin2 A a # a = a2 Two other useful forms of this identity can be obtained by substituting cos2 A = 1 - sin2 A or sin2 A = 1 - cos2 A. Replacing cos2 A with the expression 1 - sin2 A gives the following. cos 2 A = cos2 A - sin2 A Double-angle identity from above = 11 - sin2 A2 - sin2 A Fundamental identity cos 2 A =1 −2 sin2 A Subtract. Replacing sin2 A with 1 - cos2 A gives a third form. cos 2 A = cos2 A - sin2 A Double-angle identity from above = cos2 A - 11 - cos2 A2 Fundamental identity = cos2 A - 1 + cos2 A Distributive property cos 2 A =2 cos2 A −1 Add.
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