Question

How to find the value of $\cos \left({120}^{°}\right)$?

Answer 1

Find the value of $\cos \left({120}^{°}\right)$

We know that

$\cos \left(\alpha +\beta \right)=\cos \left(\alpha \right)\cos \left(\beta \right)-\sin \left(\alpha \right)\sin \left(\beta \right)$

Now by substituting the value of $\alpha ={90}^{°}$ and $\beta ={30}^{°}$in the trigonometric identity we get

$$\begin{gathered} \cos \left({90}^{°}+{30}^{°}\right)=\cos \left({90}^{°}\right)\cos \left({30}^{°}\right)-\sin \left({90}^{°}\right)\sin \left({30}^{°}\right) \\ \Rightarrow \cos \left({90}^{°}+{30}^{°}\right)=0-1\times \frac{1}{2}\left[\because \cos \left({90}^{°}\right)=0,\sin \left({90}^{°}\right)=1\mathrm{and}\sin \left({30}^{°}\right)=\frac{1}{2}\right] \\ \Rightarrow \cos \left({120}^{°}\right)=-\frac{1}{2} \end{gathered}$$.

Hence, the value of $\cos \left({120}^{°}\right)$ is $-\frac{1}{2}$.