Question

Solve the equation: cos θ - cos 7θ = sin 4θ

Answer 1

Answer:

Step-by-step explanation:

Formula used:

$cosC-cosD=-2sin(\frac{C+D}{2}) sin(\frac{C-D}{2})\\\\The\:solution\:of\:sin\theta=sin\alpha\:is\:\theta=n\pi+{(-1)}^n\alpha$

cos θ - cos 7θ = sin 4θ

$-2sin(\frac{\theta+7\theta}{2}) sin(\frac{\theta-7\theta}{2})=sin4\theta$

$-2\:sin4\theta. sin(-3\theta)=sin4\theta\\\\2\:sin4\theta. sin3\theta-sin4\theta=0\\\\sin4\theta (2sin3\theta-1)=0$

$(1)sin4\theta=0\\4\theta=n\pi\\\theta=\frac{n\pi}{4}$\\

$(2)2sin3\theta-1=0\\sin3\theta=\frac{1}{2}\\sin3\theta=sin\frac{\pi}{6}\\3\theta=n\pi+{(-1)}^n\frac{\pi}{6}\\\theta=\frac{n\pi}{3}+{(-1)}^n\frac{\pi}{18}$