Chapter 4.2, Problem 25E

To determine

To evaluate the six trigonometric functions values at $t=\frac{4\pi }{3}$ .

Answer to Problem 25E

  $\begin{array}{l}\sin \left(\frac{4\pi }{3}\right)=-\frac{\sqrt{3}}{2},\cos \left(\frac{4\pi }{3}\right)=-\frac{1}{2},\tan \left(\frac{4\pi }{3}\right)=\sqrt{3},\text{cosec}\left(\frac{4\pi }{3}\right)=-\frac{2}{\sqrt{3}},\sec \left(\frac{4\pi }{3}\right)=-2,\\ \cot \left(\frac{4\pi }{3}\right)=\frac{1}{\sqrt{3}}.\end{array}$

Explanation of Solution

Given information :

Given six trigonometric functions $\sin x,\cos x,\tan x,\cot x,\sec x,\cos ecx$ .

As we know that $\cos ecx=\frac{1}{\sin x},\sec x=\frac{1}{\cos x},\cot x=\frac{1}{\tan x}$

  $\begin{array}{l}\text{To}\text{find}\sin \left(\frac{4\pi }{3}\right)=\sin \left(\pi +\frac{\pi }{3}\right)\\ =-\sin \left(\frac{\pi }{3}\right)\left(\because \sin \left(\pi +k\right)=-\sin k\right)\\ \therefore \sin \left(\frac{4\pi }{3}\right)=-\frac{\sqrt{3}}{2}\\ \Rightarrow \cos ec\left(\frac{4\pi }{3}\right)=\frac{1}{\sin \left(\frac{4\pi }{3}\right)}=-\frac{\sqrt{3}}{2}\\ \text{similarly},\cos \left(\frac{4\pi }{3}\right)=\cos \left(\pi +\frac{\pi }{3}\right)\\ =-\cos \left(\frac{\pi }{3}\right)\left(\because \cos \left(\pi +k\right)=-\cos k\right)\\ \therefore \cos \left(\frac{4\pi }{3}\right)=-\frac{1}{2}\end{array}$

  $\begin{array}{l}\Rightarrow sec\left(\frac{4\pi }{3}\right)=\frac{1}{\cos \left(\frac{4\pi }{3}\right)}\\ \therefore sec\left(\frac{4\pi }{3}\right)=-2\\ \tan \left(\frac{4\pi }{3}\right)=\frac{\sin \left(\frac{4\pi }{3}\right)}{\cos \left(\frac{4\pi }{3}\right)}=\frac{-\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ \therefore \tan \left(\frac{4\pi }{3}\right)=\sqrt{3}\\ \cot \left(\frac{4\pi }{3}\right)=\frac{1}{\tan \left(\frac{4\pi }{3}\right)}\\ =\frac{1}{\sqrt{3}}\\ \therefore \cot \left(\frac{4\pi }{3}\right)=\frac{1}{\sqrt{3}}\end{array}$

  $\begin{array}{l}Thus,\sin \left(\frac{4\pi }{3}\right)=-\frac{\sqrt{3}}{2},\cos \left(\frac{4\pi }{3}\right)=-\frac{1}{2},\tan \left(\frac{4\pi }{3}\right)=\sqrt{3},\text{cosec}\left(\frac{4\pi }{3}\right)=-\frac{2}{\sqrt{3}},\sec \left(\frac{4\pi }{3}\right)=-2,\\ \cot \left(\frac{4\pi }{3}\right)=\frac{1}{\sqrt{3}}.\end{array}$