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.

Solution Steps

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

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

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

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

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

Final Answer