Trigonometric Equations

Trigonometric Equations - Triple Angle Formula


In the domain [0,2π] [0, 2\pi] , how many solutions are there to

cos3θ=4cosθ+3? \cos 3 \theta = 4 \cos \theta + 3 ?

If provided that sinx+cosx=16,\sin x+\cos x=\frac{1}{6}, what is the value of sin3xcos3x?\sin 3x-\cos 3x ?

Suppose that cosθ=24\displaystyle \cos \theta=\frac{\sqrt{2}}{4} for 0<θ<π2.\displaystyle 0 < \theta < \frac{\pi}{2}. Then sin3θ+cos3θ\sin 3\theta+\cos 3\theta can be expressed as sin3θ+cos3θ=a2+b148\sin 3\theta+\cos 3\theta=\frac{a\sqrt{2}+b\sqrt{14}}{8} for rational numbers aa and b.b. What is the value of 3a+8b?3a+8b ?

If provided that sinx+cosx=35,\sin x+\cos x=\frac{\sqrt{3}}{5}, where sinx>cosx,\sin x > \cos x, what is the value of sin3x+cos3x?\sin 3x+\cos 3x ?

Suppose that sin3θ=sin2θ\sin 3\theta=\sin 2\theta for 0<θ<π2.\displaystyle 0 < \theta < \frac{\pi}{2}. Then what is the value of 20cos2θ10cosθ?20\cos^2 \theta-10\cos \theta ?


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