find the exact value of the expression.\nsec sin ^ { - 1 } ( \frac { 1 } { 2 } ) \nselect the correct choice…

find the exact value of the expression.\nsec sin ^ { - 1 } ( \frac { 1 } { 2 } ) \nselect the correct choice and fill in any answer boxes in your choice below.\na. sec sin ^ { - 1 } ( \frac { 1 } { 2 } ) = □\n(type an exact answer, using radicals as needed. simplify your answer. use ir\nb. there is no solution.

find the exact value of the expression.\nsec sin ^ { - 1 } ( \frac { 1 } { 2 } ) \nselect the correct choice and fill in any answer boxes in your choice below.\na. sec sin ^ { - 1 } ( \frac { 1 } { 2 } ) = □\n(type an exact answer, using radicals as needed. simplify your answer. use ir\nb. there is no solution.

Answer

Explanation:

Step1: Let $\theta=\sin^{-1}\left(\frac{1}{2}\right)$

By the definition of the inverse - sine function, if $\theta = \sin^{-1}(x)$, then $\sin\theta=x$ and $\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. So, if $\theta=\sin^{-1}\left(\frac{1}{2}\right)$, then $\sin\theta=\frac{1}{2}$ and $\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. The value of $\theta$ for which $\sin\theta=\frac{1}{2}$ and $\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is $\theta=\frac{\pi}{6}$.

Step2: Find the value of $\sec\theta$

We know that $\sec\theta=\frac{1}{\cos\theta}$. Since $\theta = \frac{\pi}{6}$, and $\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$. Then $\sec\left(\frac{\pi}{6}\right)=\frac{1}{\cos\left(\frac{\pi}{6}\right)}$. Substitute $\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$ into the formula: $\sec\left(\frac{\pi}{6}\right)=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$.

Answer:

A. $\sec\left[\sin^{-1}\left(\frac{1}{2}\right)\right]=\frac{2\sqrt{3}}{3}$