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)(-(1)/(2)))`

To find:

The value of given expression.

Solution:

We have,

`\tan (\cos ^(-1)(-(1)/(2)))`

`=\tan (\pi-\cos^(-1)((1)/(2)))`
`[\because \cos^(-1)(-\theta)=\pi -\cos^(-1)\theta]`

`=\tan (\pi-\cos^(-1)(\cos (\pi)/(3)))`
`[\because \cos (\pi)/(3)=(1)/(2)]`

`=\tan (\pi-(\pi)/(3))`

`=-\tan ((\pi)/(3))`
`[\because \tan(\pi-\theta)=-\tan \theta]`

`=-√(3)`
`[\because \tan (\pi)/(3)=√(3)]`

Therefore, the value of given expression is
`-√(3)`.