The number of solutions of the equation 2x-2 x+4 x=4 in the interval [0,5] is:

Question

The number of solutions of the equation $\sin2x-2\cos x+4\sin x=4$ in the interval $[0,5\pi]$ is:

Accepted Answer

$$2\sin x\cos x - 2\cos x+4\sin x-4=0 \implies (\sin x-1)(2\cos x)... $$

Grouping: $2\cos x(\sin x-1) + 4(\sin x-1) = 0 \implies (\sin x-1)(2\cos x+4) = 0$

Since $\cos x+2\ne0$ always, $\sin x=1 \implies x = \dfrac\pi2,\dfrac{5\pi}2,\dfrac{9\pi}2$ --- 3 solutions in $[0,5\pi]$.

Solutions · 1