Find the exact value of the expression.

Select the correct choice and fill in any answer boxes in your choice below.
A.
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
B. There is no solution.

Ask by Valdez Harris. in the United States

Jan 19,2025

Question

Find the exact value of the expression.

Select the correct choice and fill in any answer boxes in your choice below.
A.
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
B. There is no solution.

Ask by Valdez Harris. in the United States

Jan 19,2025

UpStudy AI · Accepted Answer

To find the exact value of $$ \tan \left[\sin^{-1}\left(\frac{4}{9}\right)\right] $$, let's follow these steps:

1. **Let $$ \theta = \sin^{-1}\left(\frac{4}{9}\right) $$.**
   - This means $$ \sin(\theta) = \frac{4}{9} $$.

2. **Use the Pythagorean identity to find $$ \cos(\theta) $$.**
   $$
   \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{4}{9}\right)^2} = \sqrt{1 - \frac{16}{81}} = \sqrt{\frac{65}{81}} = \frac{\sqrt{65}}{9}
   $$

3. **Calculate $$ \tan(\theta) $$.**
   $$
   \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{4}{9}}{\frac{\sqrt{65}}{9}} = \frac{4}{\sqrt{65}}
   $$

4. **Rationalize the denominator.**
   $$
   \frac{4}{\sqrt{65}} \times \frac{\sqrt{65}}{\sqrt{65}} = \frac{4\sqrt{65}}{65}
   $$

So, the exact value is:

$$
\tan \left[\sin^{-1}\left(\frac{4}{9}\right)\right] = \frac{4\sqrt{65}}{65}
$$

**Answer:**

**A.** $$ \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right]=\dfrac{4\,\sqrt{65}}{65} $$
$$ \tan \left[\sin^{-1}\left(\frac{4}{9}\right)\right] = \frac{4\sqrt{65}}{65} $$

Find the exact value of the expression.

Select the correct choice and fill in any answer boxes in your choice below.
A.
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
B. There is no solution.

Upstudy AI Solution

Answer

Solution

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Find the exact value of the expression. Select the correct choice and fill in any answer boxes in your choice below. A. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. There is no solution.

Upstudy AI Solution

Answer

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Find the exact value of the expression.

Select the correct choice and fill in any answer boxes in your choice below.
A.
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
B. There is no solution.