In this section we have seen that the cosines of the

Chapter 9, Problem 56

(choose chapter or problem)

In this section we have seen that the cosines of the angles in a triangle can be expressed in terms of the lengths of the sides. For instance, for cos A in ^ABC, we obtained cos A (b2 c2 a2 )/2bc. This exercise shows how to derive corresponding expressions for the sines of the angles. For ease of notation in this exercise, let us agree to use the letter T to denote the following quantity: T 2(a2 b2 b2 c2 c2 a2 ) (a4 b4 c4 ) Then the sines of the angles in ^ABC are given by In the steps that follow, well derive the first of these three formulas, the derivations for the other two being entirely similar. (a) In ^ABC, why is the positive root always appropriate in the formula sin A (b) In the formula in part (a), replace cos A by (b2 c2 a2 )/2bc and show that the result can be written (c) On the right-hand side of the equation in part (b), carry out the indicated multiplication. After combining like terms, you should obtain sin A /2bc, as required. 57.