Here I have this trigonometric equation $\cos 7\theta=\cos 3\theta+\sin 5\theta$.
I approached the problem as follows :
$\cos 7\theta=\cos 3\theta+\sin 5\theta$
$\implies \cos 7\theta-\cos 3\theta=\sin 5\theta$
$\implies 2\sin 5\theta\sin(-2\theta)=\sin 5\theta$
From there on we get,
$\sin 2\theta=-\dfrac{1}{2}$ and $\sin 5\theta=0$
So, we deduce that $\theta=-\dfrac{\pi}{12}+\pi k$ and $\theta=\dfrac{2}{5}\pi n$ where $k$ and $n$ are integers.
However, Wolfram|Alpha doesn't seem to agree with me. They present $\dfrac{\pi}{5}$ as a solution which is not included in my solution.
Can someone help me out?
Best Answer
$$\sin 5\theta=0$$ $\sin x =0$ has solutions $x=n\pi$
Thus we get $$5\theta=n \pi$$
$$\theta= \frac{n\pi}{5}\hspace{10pt} n\in Z$$
Also $$\sin 2\theta=-\dfrac{1}{2}$$
$$2\theta=2n\pi-\frac{\pi}{6} \implies \theta=n\pi-\frac{\pi}{12}\hspace{10pt} n\in Z$$ $$2\theta=(2n+1)\pi+\frac{\pi}{6} \implies \theta=n\pi+\frac{7\pi}{12}\hspace{10pt} n\in Z$$