Question

If ${\sin }^{2}x-\cos x=1/4$, then the value of $x$ between $0$ and $2\pi$ are :

• A. $\pi /3,5\pi /3$
• B. $\pi /3,-\pi /3$
• C. $2\pi /3,\pi /3$
• D. $Noneofthese$

Answer 1

Answer: (A)

$\pi /3,5\pi /3$

Given

${\sin }^{2}x-\cos x=\frac{1}{4}$

$⟹1-{\cos }^{2}x-\cos x=\frac{1}{4}$

$⟹4-4{\cos }^{2}x-4\cos x=1$

$⟹4{\cos }^{2}x+4\cos x-3=0$

$⟹4{\cos }^{2}x+6\cos x-2\cos x-3=0$

$⟹2\cos x(2\cos x+3)-1(2\cos x+3)=0$

$⟹(2\cos x-1)(2\cos x+3)=0$

$⟹\cos x=\frac{1}{2}$ and $\cos x\ne -\frac{3}{2}$

$⟹x=\frac{\pi }{3},\frac{5\pi }{3}$