# Math Help - New Trigo-equation

1. ## New Trigo-equation

Solve in R : $\sqrt{-3-\cos ^{2}(x)+3\sin (5x)}=1-\sin (x)$

2. Condition: $-3-\cos^2x+3\sin 5x\geq 0$

But $3(\sin 5x-1)\leq 0$ and $-\cos^2x\leq 0$.

Then $\sin 5x=1$ and $\cos x=0$.

Also the right side member must be 0: $1-\sin x=0$

$\sin 5x=16\sin^5x-20\sin^3x+5\sin x$

If $\sin x=1$ then $\sin 5x=1$.

So $\sin x=1\Rightarrow x=\frac{\pi}{2}+2k\pi$

3. from red_dog: ,

when k = 0, then x = pi/2,

k = 1, x = 7pi/2.

The graph confirms the rightfullness of red_dog's Solution.

ok