To solve the expression [tex]\(\tan \left( \sin^{-1} \frac{1}{3} \right)\)[/tex], we can follow these steps:
1. Identify the angle [tex]\(\theta\)[/tex]:
Let's denote the angle by [tex]\(\theta\)[/tex] such that [tex]\(\theta = \sin^{-1}\left( \frac{1}{3} \right)\)[/tex].
This means that [tex]\(\sin \theta = \frac{1}{3}\)[/tex].
2. Find [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] whose sine value is [tex]\(\frac{1}{3}\)[/tex] is approximately [tex]\(0.3398369094541219\)[/tex] radians.
3. Calculate [tex]\(\tan \theta\)[/tex]:
Use the values of trigonometric functions for the angle [tex]\(\theta\)[/tex].
The tangent of [tex]\(\theta\)[/tex] can be found:
[tex]\[
\tan \theta \approx 0.35355339059327373
\][/tex]
Next, we look at the provided choices to see which one matches the value [tex]\(0.35355339059327373\)[/tex]:
- A: [tex]\(\frac{\sqrt{2}}{4} \approx 0.3535533905932738\)[/tex]
The correct answer is [tex]\(A\)[/tex] [tex]\(\frac{\sqrt{2}}{4}\)[/tex], as it matches the calculated value of the tangent of the given angle.
Thus, the solution is:
[tex]\[
\tan \left( \sin^{-1} \frac{1}{3} \right) = \frac{\sqrt{2}}{4}
\][/tex]
So, the correct answer is option [tex]\(A\)[/tex].
