Question

Find the angle between $\pi$ and $2\pi$ that satisfies the equation: $2{\sin }^{2}x+5\cos x+1=0$

• A. $\frac{7\pi }{6}$
• B. $\frac{3\pi }{2}$
• C. $\frac{5\pi }{3}$
• D. $\frac{11\pi }{6}$
• E. $\frac{4\pi }{3}$

Answer 1

The correct option is E $\frac{4\pi }{3}$

Given, $2{\sin }^{2}x+5\cos x+1=0$

• $\Rightarrow 2-2{\cos }^{2}x+5\cos x+1=0$
• $\Rightarrow 2{\cos }^{2}x-5\cos x-3=0$

We get $\cos x=\frac{(5\pm \sqrt{25+24})}{4}=\frac{(5\pm \sqrt{49})}{4}=\frac{(5\pm 7)}{4}=3,-\frac{1}{2}$
Therefore $\cos x=-\frac{1}{2}$ ($3$ is not in range of $\cos x$)
We get $x=240=\frac{4\pi }{3}$