Question

How do you evaluate $\cos \left(\frac{4\mathrm{\pi }}{3}\right)$?

Answer 1

Find the value of $\cos \left(\frac{4\mathrm{\pi }}{3}\right)$:

$$\begin{gathered} \Rightarrow \cos \left(\frac{4\mathrm{\pi }}{3}\right)=\cos \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right) \\ \Rightarrow \cos \left(\frac{4\mathrm{\pi }}{3}\right)=-\cos \left(\frac{\mathrm{\pi }}{3}\right)\left[\because \cos \left(\mathrm{\pi }+\theta \right)=-\cos \left(\theta \right)\right] \\ \Rightarrow \cos \left(\frac{4\mathrm{\pi }}{3}\right)=-\left(\frac{1}{2}\right)\left[\because \cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2}\right] \\ \therefore \cos \left(\frac{4\mathrm{\pi }}{3}\right)=-\frac{1}{2} \end{gathered}$$

Hence, the value of $\cos \left(\frac{4\mathrm{\pi }}{3}\right)$ is $-\frac{1}{2}$.