536 CHAPTER 7 Analytic Trigonometry Solution (a) Use formula (1) to get sin6 sin4 1 2 cos6 4 cos6 4 1 2 cos 2 cos 10 θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] = − − + = − (b) Use formula (2) to get cos3 cos 1 2 cos 3 cos 3 1 2 cos 2 cos 4 θ θ θ θ θ θ θ θ ( ) ( ) ( ) [ ] ( ) ( ) [ ] = − + + = + (c) Use formula (3) to get sin3 cos5 1 2 sin3 5 sin3 5 1 2 sin 8 sin 2 1 2 sin 8 sin 2 θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] = + + − = + − = − Expressing Sums (or Differences) as Products Express each sum or difference as a product of sines and/or cosines. (a) sin 5 sin 3 θ θ ( ) ( ) − (b) cos 3 cos 2 θ θ ( ) ( ) + EXAMPLE 2 Now Work PROBLEM 7 2 Express Sums as Products THEOREM Sum-to-Product Formulas • sin sin 2 sin 2 cos 2 α β α β α β + = + − (6) • sin sin 2 sin 2 cos 2 α β α β α β − = − + (7) • cos cos 2cos 2 cos 2 α β α β α β + = + − (8) • cos cos 2 sin 2 sin 2 α β α β α β − = − + − (9) Formula (6) is derived here. The derivations of formulas (7) through (9) are left as exercises (see Problems 56 through 58). Proof 2 sin 2 cos 2 2 1 2 sin 2 2 sin 2 2 sin 2 2 sin 2 2 sin sin α β α β α β α β α β α β α β α β ( ) ( ) + − = ⋅ + + − + + − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + = + ■ ↑ Product-to-Sum Formula (3)
RkJQdWJsaXNoZXIy NjM5ODQ=