Section 7.7 AN59 25. θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) + = = sin sin 3 2 sin 2 2 sin 2 cos 2 sin 2 cos 27. θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = = = sin 4 sin 2 cos 4 cos 2 2 sin 3 cos 2 cos 3 cos sin 3 cos 3 tan 3 29. θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) − + = = = cos cos 3 sin sin 3 2 sin 2 sin 2 sin 2 cos sin cos tan 31. θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] + = = = ⋅ − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − sin sin sin 3 sin 2 sin 2 cos cos 2 sin 2 sin cos 2 1 2 cos cos 3 cos cos cos 3 33. θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = = = sin 4 sin 8 cos 4 cos 8 2sin 6 cos 2 2cos 6 cos 2 sin 6 cos 6 tan 6 35. θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) + − = − − = ⋅ − = − = − sin 4 sin 8 sin 4 sin 8 2sin 6 cos 2 2sin 2 cos 6 sin 6 cos 6 cos 2 sin 2 tan 6 cot 2 tan 6 tan 2 37. α β α β α β α β α β α β α β α β α β α β α β α β + − = + − − + = + + ⋅ − − = + − sin sin sin sin 2 sin 2 cos 2 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 tan 2 cot 2 39. α β α β α β α β α β α β α β α β α β + + = + − + − = + + = + sin sin cos cos 2 sin 2 cos 2 2 cos 2 cos 2 sin 2 cos 2 tan 2 41. θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) + + + = + + + = + − 1 cos2 cos4 cos6 1 cos 6 cos 2 cos 4 2 cos 3 2 cos 3 cos 2 θ θ θ θ θ θ θ θ θ ( )[ ( ) ] ( ) ( ) [ ] ( ) ( ) = + = = 2 cos 3 cos 3 cos 2 cos 3 2 cos 2 cos 4 cos cos 2 cos 3 43. θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) [ ] ( ) [ ] = = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ⋅ + = − + sin cos sin cos 1 cos 2 2 1 cos 2 2 1 8 1 cos2 1 cos2 4 2 2 2 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ [ ] ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) [ ] = − − = − − ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = − − + = − − = − − + 1 8 1 cos2 1 cos 2 1 8 1 cos 2 1 1 cos 4 2 1 16 1 cos2 2 1 cos4 1 16 1 cos2 1 cos4 1 16 1 cos2 cos4 cos4 cos2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ { } ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) = − − + + = − − + + = − − + = − − + 1 16 1 cos 2 cos 4 1 2 cos 2 cos 6 1 32 2 2cos 2 2cos 4 cos 2 cos 6 1 32 2 cos 2 2cos 4 cos 6 1 16 1 32 cos 2 1 16 cos 4 1 32 cos 6 45. θ θ θ θ ( ) ( ) ( ) [ ] = = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − sin sin 1 cos 2 2 1 8 1 cos 2 6 2 3 3 3 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ [ ] [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) = − + − = − + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − = − + + − = − + − = − − + + − 1 8 1 2cos2 cos 2 1 cos2 1 8 1 2 cos 2 1 cos 4 2 1 cos 2 1 16 2 4cos2 1 cos4 1 cos2 1 16 3 4cos2 cos4 1 cos2 1 16 3 3cos 2 4cos 2 4cos 2 cos 4 cos 4 cos 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ { } ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) = − + ⋅ + + − + = − + + + − − = − + − = − + − 1 16 3 7 cos 2 4 1 cos 4 2 cos 4 1 2 cos 2 cos 6 1 32 6 14cos 2 4 4cos 4 2cos 4 cos 2 cos 6 1 32 10 15 cos 2 6 cos 4 cos 6 5 16 15 32 cos 2 3 16 cos 4 1 32 cos 6 47. π π π π π π π { } 0, 3 , 2 , 2 3 , , 4 3 , 3 2 , 5 3 49. π π π π π π π π π { } 0, 5 , 2 5 , 3 5 , 4 5 , , 6 5 , 7 5 , 8 5 , 9 5 51. (a) π π ( ) ( ) = y t t 2 sin 2061 cos 357 (b) = y 2 max (c) 2 22 0.01 0 53. θ θ θ θ θ θ θ = + − = + − I I I I I I I cos sin 2 sin cos cos sin sin2 u x y xy x y xy 2 2 2 2 θ θ θ θ θ θ θ θ ( ) ( ) = + + − − = + + − − = + + − − I I I I I I I I I I I I I cos2 1 2 1 cos2 2 sin2 2 cos2 2 2 2 cos2 sin2 2 2 cos2 sin2 x y xy x x y y xy x y x y xy θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) = + + = − + + + = − + + + = + − − + I I I I I I I I I I I I I I I I I sin cos 2 sin cos 1 cos2 2 cos2 1 2 sin2 2 2 cos2 2 cos2 2 sin2 2 2 cos2 sin2 v x y xy x y xy x x y y xy x y x y xy 2 2 55. α β α β α β α β α β α β α β α β α β α β α β α β ( ) ( ) ( ) ( ) ( ) ( ) [ ] − = − + = + − + + = = + + − sin sin cos cos sin sin sin cos cos sin sin sin 2 sin cos sin cos 1 2 sin sin 57. α β α β α β α β α β α β α β α β ( ) ( ) + − = ⋅ + + − + + − − ⎡ ⎢ ⎣ ⎤ ⎦ ⎥ = + = + 2 cos 2 cos 2 2 1 2 cos 2 2 cos 2 2 cos 2 2 cos 2 2 cos cos
RkJQdWJsaXNoZXIy NjM5ODQ=