Question

Solve the trignometric equation and find ALL solutions
$f\left(x\right)=2\cos x+1=0$

• A. $\frac{\pi }{3}+2n\pi ,\frac{5\pi }{3}+2n\pi$
• B. $-1/2$
• C. no solutions
• D. $\frac{2\pi }{3}+2n\pi ,\frac{4\pi }{3}+2n\pi$

Answer 1

The correct option is D $\frac{2\pi }{3}+2n\pi ,\frac{4\pi }{3}+2n\pi$

$f\left(x\right)=2\cos x+1=0$
$2\cos x=-1$
$\cos x=-\frac{1}{2}$
Value of $\cos x$ is negative in 2nd and 3rd quadrant
$\cos (\pi -\frac{\pi }{3})=\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$
$\cos (\pi +\frac{\pi }{3})=\cos \left(\frac{4\pi }{3}\right)=-\frac{1}{2}$
$x=\frac{2\pi }{3}+2n\pi ,\frac{4\pi }{3}+2n\pi$