Question

Find the value of ${{\sec }^{-1}}[\sec (-{{30}^{\circ }})]$ .

Answer 1

Hint: According to definition of inverse sec function we can write
${{\sec }^{-1}}[\sec ({{x}^{\circ }})]={{x}^{\circ }}$ if $x\in \left[ 0,\dfrac{\pi }{2} \right)\cup \left( \dfrac{\pi }{2},\pi \right]$

Complete step-by-step answer:
Given expression is ${{\sec }^{-1}}[\sec (-{{30}^{\circ }})]$
According to definition of inverse sec function ${{\sec }^{-1}}[\sec ({{x}^{\circ }})]={{x}^{\circ }}$ if $x\in \left[ 0,\dfrac{\pi }{2} \right)\cup \left( \dfrac{\pi }{2},\pi \right]$
But $-{{30}^{\circ }}$ is not in the domain of inverse sec function.
We can use $\sec \left( -\theta \right)=\sec \left( \theta \right)$.
So we can write $\sec \left( -{{30}^{\circ }} \right)=\sec \left( {{30}^{\circ }} \right)$
Hence we can write given expression as ${{\sec }^{-1}}[\sec (-{{30}^{\circ }})]={{\sec }^{-1}}[\sec ({{30}^{\circ }})]$
Now we can simplify it as
${{\sec }^{-1}}[\sec ({{30}^{\circ }})]={{30}^{\circ }}$ because ${{30}^{\circ }}$ is in domain of inverse sec function.

Note: In this type of function we need to first check that angle is in the domain of inverse function or not. If angle is not in domain we need to first convert to write it in domain of inverse trigonometric function.