Question

The number of solutions of the equation $\sin x=\cos 3x$ in $[0,\pi ]$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

The given equation can be written as

$\sin x=4{\cos }^{3}x-3\cos x\Rightarrow {\sec }^{2}x\tan x+3{\sec }^{2}x-4=0$

In terms of $\tan x$, this leads to the equation

${\tan }^{3}x+3{\tan }^{2}x+\tan x-1=0\Rightarrow (\tan x+1)({\tan }^{2}x+2\tan x-1)=0$

$\Rightarrow \tan x=-1$ or $\tan 2x=1$

$\Rightarrow x=\frac{3\pi }{4},\frac{\pi }{8},\frac{5\pi }{8}$