To find the exact value of the expression [tex]\(\csc \left(\sin^{-1}\left(\frac{1}{2}\right)\right)\)[/tex], we can go through the following steps:
1. Identify the angle [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \sin^{-1}\left(\frac{1}{2}\right)
\][/tex]
This represents the angle whose sine value is [tex]\(\frac{1}{2}\)[/tex].
2. Find the angle [tex]\(\theta\)[/tex]:
We know from basic trigonometry that [tex]\(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)[/tex]. Therefore,
[tex]\[
\theta = \frac{\pi}{6}
\][/tex]
3. Express the given expression in terms of [tex]\( \theta \)[/tex]:
Substitute [tex]\(\theta = \frac{\pi}{6}\)[/tex] into the given expression:
[tex]\[
\csc \left(\sin^{-1}\left(\frac{1}{2}\right)\right) = \csc\left(\frac{\pi}{6}\right)
\][/tex]
4. Recall the definition of cosecant:
The cosecant of an angle is the reciprocal of its sine. Therefore,
[tex]\[
\csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)}
\][/tex]
5. Substitute the sine value:
We already know that [tex]\(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)[/tex]. So,
[tex]\[
\csc\left(\frac{\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2
\][/tex]
Therefore, the exact value of the expression [tex]\(\csc \left(\sin^{-1}\left(\frac{1}{2}\right)\right)\)[/tex] is [tex]\(2\)[/tex]. The intermediate value of the angle [tex]\(\sin^{-1}\left(\frac{1}{2}\right)\)[/tex] is [tex]\(\frac{\pi}{6}\)[/tex].
