cos³θ + sin³θ = (1/4)(cos³θ +3cosθ - sin³θ + 3sinθ)
cos3θ + sin3θ = (1/4)(cos3θ +3cosθ - sin3θ + 3sinθ)
(1/4)(3cosθ + cos3θ + 3sinθ - sin3θ) = (1/4)(cos3θ +3cosθ - sin3θ + 3sinθ)
3cosθ + cos3θ + 3sinθ - sin3θ = cos3θ +3cosθ - sin3θ + 3sinθ
cos3θ - sin3θ = cos3θ - sin3θ
cos3θ - sin3θ =(1/4)(3cosθ + cos3θ - 3sinθ - sin3θ)
4cos3θ - 4sin3θ = 3cosθ + cos3θ - 3sinθ - sin3θ
-3cosθ + 3cos3θ + 3sinθ - 5sin3θ = 0
-2sinθ(1 + 5cos2θ + 3sin2θ) = 0
sinθ = 0 or 1 + 5cos2θ + 3sin2θ = 0
θ = n1 for n1 Z
substitute y = tanθ.
1 + 5/(y2 + 1) + 6y/(y2 + 1) - 5y2/(y2 + 1) = 0
(2(2y2 - 3y - 3))/(y2 + 1) = 0
2y2 - 3y - 3 = 0
tanθ = 3/4 - /4 or tanθ = 3/4 + /4
θ = tan-1(3/4 - /4) + n2 for n2 Z
θ = tan-1(3/4 + /4) + n3 for n3 Z
Answer:
θ = n1 for n1 Z
θ = tan-1(3/4 - /4) + n2 for n2 Z
θ = tan-1(3/4 + /4) + n3 for n3 Z
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