Question

How do you find the six trigonometric functions of $\left(\frac{-2\mathrm{\pi }}{3}\right)$ degrees?

Answer 1

Step-1 : Compute the value of $\sin \theta$ and $cosec\theta$.

The angle $\frac{-2\mathrm{\pi }}{3}$ can be written as $\left(-\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)$.

The value of $\sin \left(-\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)$ can be written as $-\sin \left(\frac{\mathrm{\pi }}{3}\right)$.

The value of $-\sin \left(\frac{\mathrm{\pi }}{3}\right)$ is $\frac{-\sqrt{3}}{2}$.

Therefore, $\sin \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{\sqrt{3}}{2}$

The value of $cosec\theta$ is $\frac{1}{\sin \theta }$.

The value of is $cosec\left(\frac{-2\mathrm{\pi }}{3}\right)=\frac{-2}{\sqrt{3}}$.

Step-2 : Compute the value of $\cos \theta$ and $\sec \theta$.

The value of $\cos \left(-\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)$ can be written as $-\cos \left(\frac{\mathrm{\pi }}{3}\right)$.

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

Therefore, $\cos \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{1}{2}$

The value of $\sec \theta$ is $\frac{1}{\cos \theta }$.

The value of $\sec \left(\frac{-2\mathrm{\pi }}{3}\right)=-2$.

Step-3: Compute the value of $\tan \theta$ and $\cot \theta$.

The value of $\tan \theta$ is $\frac{\sin \theta }{\cos \theta }$.

Divide $\sin \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{\sqrt{3}}{2}$by $\cos \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{1}{2}$to obtain the value of $\tan \left(\frac{-2\mathrm{\pi }}{3}\right)$.

$$\begin{gathered} \tan \left(\frac{-2\mathrm{\pi }}{3}\right)=\frac{{\displaystyle \frac{-\sqrt{3}}{2}}}{{\displaystyle \frac{-1}{2}}} \\ \therefore \tan \left(\frac{-2\mathrm{\pi }}{3}\right)=\sqrt{3} \end{gathered}$$

The value of $\cot \theta$ is $\frac{1}{\tan \theta }$.

The value of $\cot \left(\frac{-2\mathrm{\pi }}{3}\right)=\frac{1}{\sqrt{3}}$.

Hence, the value of $\sin \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{\sqrt{3}}{2}$, $\cos \left(\frac{-2\mathrm{\pi }}{3}\right)=-\frac{1}{2}$, $\tan \left(\frac{-2\mathrm{\pi }}{3}\right)=\sqrt{3}$ , $cosec\left(\frac{-2\mathrm{\pi }}{3}\right)=\frac{-2}{\sqrt{3}}$, $\sec \left(\frac{-2\mathrm{\pi }}{3}\right)=-2$,$\cot \left(\frac{-2\mathrm{\pi }}{3}\right)=\frac{1}{\sqrt{3}}$.