Question

Solve the equation
$\left(i\right){\sin }^{2}x-\cos x=\frac{1}{4}$

Answer 1

Given: ${\sin }^{2}x-\cos x=\frac{1}{4}$
$\Rightarrow 1-{\cos }^{2}x-\cos x=\frac{1}{4}\Rightarrow 4{\cos }^{2}x+4\cos x-3=0\Rightarrow (2\cos x-1)(2\cos x+3)=0\Rightarrow (2\cos x-1)=0\text{or}(2\cos x+3)=0\Rightarrow \cos x=\frac{1}{2}\text{or}\cos x=-\frac{3}{2}$

As $\cos x\in [-1,1]$ so $\cos x=\frac{1}{2}$
$\Rightarrow \cos x=\cos \frac{\pi }{3}$

We know the geneal solution of $\cos x=\cos \alpha$ is
$x=2n\pi \pm \alpha ,n\in Z$

Hence, the general solution is
$x=2n\pi \pm \frac{\pi }{3},n\in Z$