JEE Main 2013 (Online) 23rd AprilQuestion 1
The number of solutions of the equation $\sin2x-2\cos x+4\sin x=4$ in the interval $[0,5\pi]$ is :
Trigonometric Ratio & Identities
Solution
Concept. Group the terms so that $\sin x-1$ comes out as a common factor. Formulas used. (1) $\sin2x=2\sin x\cos x$Step 1 - Factorise. By (1), $$\sin2x-2\cos x+4\sin x-4=2\cos x(\sin x-1)+4(\sin x-1)=(\sin x-1)(2\cos x+4).$$ Step 2 - Which factor is zero? $2\cos x+4\ge2>0$, so $\sin x=1$. Step 3 - Count in $[0,5\pi]$. $x=\frac\pi2,\ \frac{5\pi}2,\ \frac{9\pi}2$: $3$ solutions. Answer: (A)