In this exercise we will use the accompanying figure to

Chapter 8, Problem 96

(choose chapter or problem)

In this exercise we will use the accompanying figure to derive the half-angle formula for sine: Our derivation will make use of the formula for the distance between two points and the identity sin2 t cos2 t 1. However, we will not rely on an addition formula for sine, as we did in Section 8.2. (a) Explain why the coordinates of P and Q are and Q(cos u, sin u). (b) Use the distance formula to show that and (c) Explain why ^POR is congruent to ^QOP. (d) From part (c), it follows that (PQ) 2 (PR) 2 . By equating the expressions for (PQ) 2 and (PR) 2 [obtained in part (b)], show that (e) Square both sides of the equation obtained in part (d); then replace cos2 (u/2) by 1 sin2 (u/2) and show that the resulting equation can be written 3sin2 (u/2)4(2 2 cos u) (1 cos u) 2(f) Solve the equation in part (e) for the quantityYou should obtainas required