Question

If ${\sin }^{2}x-\cos x=1/4$, if the sum of values of $x$ between $0$ and $2\pi$ is $k\pi$ then find $k$.

Answer 1

Answer: 2.00

${\sin }^{2}x-\cos x=\frac{1}{4}$
$\Rightarrow 1-{\cos }^{2}x-\cos x=\frac{1}{4}$
$\Rightarrow 4{\cos }^{+2}x+y\cos x-3=0$
$\Rightarrow (2\cos x+3)(2\cos x-1)=0$
$\Rightarrow 2\cos x=1(\because \cos x\ne -3/2)$
$\Rightarrow \cos x=1/2$
$\Rightarrow \cos x=\cos \pi /3$
$\Rightarrow x=2n\pi \frac{\pm \pi }{3},n\in I$
$\therefore x=\frac{\pi }{3},\frac{5\pi }{3}\in [0,2\pi ]$