526 CHAPTER 7 Analytic Trigonometry Solution (a) In the Sum Formula for α β ( ) + tan , let α β θ = = . Then α β α β α β θ θ θ θ θ θ ( ) ( ) + = + − + = + − tan tan tan 1 tan tan tan tan tan 1 tan tan θ θ θ ( ) = − tan 2 2 tan 1 tan2 (5) θ θ ( ) = − sin 1 cos 2 2 2 (6) θ θ ( ) = + cos 1 cos 2 2 2 (7) (b) To find a formula for θ ( ) sin 3 , write θ3 as θ θ + 2 , and use the Sum Formula. θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) = + = + sin 3 sin 2 sin2 cos cos2 sin Now use the Double-angle Formulas to get θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( )( ) ( ) = + − = + − = − sin 3 2 sin cos cos cos sin sin 2 sin cos sin cos sin 3 sin cos sin 2 2 2 2 3 2 3 The formula obtained in Example 2(b) also can be written as θ θ θ θ θ θ θ θ θ ( ) ( ) = − = − − = − sin 3 3 sin cos sin 3 sin 1 sin sin 3 sin 4 sin 2 3 2 3 3 That is, θ ( ) sin 3 is a third-degree polynomial in the variable θ sin . In fact, θ ( ) n sin , n a positive odd integer, can always be written as a polynomial of degree n in the variable θ sin . * Now Work PROBLEMS 9(e) AND 69 Rearranging the Double-angle Formulas (3) and (4) leads to other formulas that are used later and are important in calculus. Begin with Double-angle Formula (3) and solve for θ sin . 2 θ θ θ θ ( ) ( ) = − = − cos 2 1 2 sin 2 sin 1 cos 2 2 2 Similarly, using Double-angle Formula (4), solve for θ cos . 2 θ θ θ θ ( ) ( ) = − = + cos 2 2 cos 1 2 cos 1 cos 2 2 2 Formulas (6) and (7) can be used to develop a formula for θ tan . 2 θ θ θ θ θ ( ) ( ) = = − + tan sin cos 1 cos 2 2 1 cos 2 2 2 2 2 *Because of the work done by P. L. Chebyshëv, these polynomials are sometimes called Chebyshëv polynomials .
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