Question

Evaluate the following trigonometric expression, using the principal value for the cosine

tan(Cos-1(-1/2))

Answer 1

Given:

The expression is

$\tan (\cos ^{-1}(-\dfrac{1}{2}))$

To find:

The value of given expression.

Solution:

We have,

$\tan (\cos ^{-1}(-\dfrac{1}{2}))$

$=\tan (\pi-\cos^{-1}(\dfrac{1}{2}))$          $[\because \cos^{-1}(-\theta)=\pi -\cos^{-1}\theta]$

$=\tan (\pi-\cos^{-1}(\cos \dfrac{\pi}{3}))$      $[\because \cos \dfrac{\pi}{3}=\dfrac{1}{2}]$

$=\tan (\pi-\dfrac{\pi}{3})$

$=-\tan (\dfrac{\pi}{3})$                 $[\because \tan(\pi-\theta)=-\tan \theta]$

$=-\sqrt{3}$         $[\because \tan \dfrac{\pi}{3}=\sqrt{3}]$

Therefore, the value of given expression is $-\sqrt{3}$.