Questions: Find the exact value of the expression. tan[sin^(-1)(9/10)] Select the correct choice and fill in any answer boxes in your choice A. tan[sin^(-1)(9/10)]= □ (Simplify your answer, including any radicals. Use integers or B. There is no solution.

Find the exact value of the expression.
tan[sin^(-1)(9/10)]

Select the correct choice and fill in any answer boxes in your choice
A. tan[sin^(-1)(9/10)]= □
(Simplify your answer, including any radicals. Use integers or
B. There is no solution.

Transcript text: Find the exact value of the expression. \[ \tan \left[\sin ^{-1}\left(\frac{9}{10}\right)\right] \] Select the correct choice and fill in any answer boxes in your choice A. $\tan \left[\sin ^{-1}\left(\frac{9}{10}\right)\right]=$ $\square$ (Simplify your answer, including any radicals. Use integers or B. There is no solution.

Solution

Solution Steps

To find the exact value of $\tan \left[\sin ^{-1}\left(\frac{9}{10}\right)\right]$, we can use the following approach:

Recognize that $\sin^{-1}\left(\frac{9}{10}\right)$ represents an angle $\theta$ such that $\sin(\theta) = \frac{9}{10}$.
Use the Pythagorean identity to find $\cos(\theta)$: $\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$.
Calculate $\tan(\theta)$ using the identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Step 1: Determine $\sin(\theta)$

Given that $\theta = \sin^{-1}\left(\frac{9}{10}\right)$, we have: \[ \sin(\theta) = \frac{9}{10} = 0.9 \]

Step 2: Calculate $\cos(\theta)$

Using the Pythagorean identity, we find $\cos(\theta)$: \[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{9}{10}\right)^2} = \sqrt{1 - \frac{81}{100}} = \sqrt{\frac{19}{100}} = \frac{\sqrt{19}}{10} \approx 0.4359 \]

Step 3: Calculate $\tan(\theta)$

Now, we can calculate $\tan(\theta)$ using the definition: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{9}{10}}{\frac{\sqrt{19}}{10}} = \frac{9}{\sqrt{19}} \approx 2.0647 \]