https://file.aiquickdraw.com/mathbot/file/17288353085558swjvvxt.jpg

The expression in the image is:

\[
\tan\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)
\]

### Step 1: Solve \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

\(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) means we are looking for the angle \(\theta\) such that:

\[
\sin(\theta) = \frac{\sqrt{3}}{2}
\]

From trigonometric values, we know that \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\). Therefore,

\[
\theta = \frac{\pi}{3}
\]

### Step 2: Apply the \(\tan\) function

Now we need to find:

\[
\tan\left(\frac{\pi}{3}\right)
\]

From standard trigonometric values, we know that:

\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]

### Final Answer:

The exact value of the given expression is:

\[
\sqrt{3}
\]

Would you like more details or further explanations?

Here are some related questions:
1. What is the definition of the inverse sine function?
2. How are trigonometric values derived from the unit circle?
3. What are other common values for \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\)?
4. How can you verify trigonometric values using a calculator?
5. What are some common applications of inverse trigonometric functions in geometry?

**Tip:** It helps to memorize the key angles (like \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\)) and their sine, cosine, and tangent values for quick reference!

Find the exact value of tan(sin⁻¹(√3/2)).

Find the exact value of tan(sin⁻¹(√3/2)) using trigonometric principles. This problem involves the unit circle, inverse sine, and tangent functions. Suitable for students in Grades 10-12.

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