Question

If ​$\cos x=-\frac{1}{2}$ and $0<x<2\mathrm{\pi },$ then the solutions are


(a) $x=\frac{\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}$

(b) $x=\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}$

(c) ​$x=\frac{2\mathrm{\pi }}{3},\frac{7\mathrm{\pi }}{6}$

(d) $\mathrm{\theta }=\frac{2\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3}$

Answer 1

(b) $x=\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}$

Given equation:
$$\begin{gathered} \cos x=-\frac{1}{2} \\ \Rightarrow \cos x=\cos \frac{2\mathrm{\pi }}{3} \\ \Rightarrow x=\frac{2\mathrm{\pi }}{3} \end{gathered}$$
Or,
$$\begin{gathered} \cos x=\cos \frac{4\mathrm{\pi }}{3} \\ \Rightarrow x=\frac{4\mathrm{\pi }}{3} \end{gathered}$$

So, both $\frac{2\mathrm{\pi }}{3}\mathrm{and}\frac{4\mathrm{\pi }}{3}$ lie in $0<x<2\mathrm{\pi }$.