https://file.aiquickdraw.com/mathbot/file/1728265515607iaw8uufk.jpg

Let's solve the inverse trigonometric expressions given in the image:

### (a) \( \sin^{-1}\left( -\frac{1}{2} \right) \)

This is asking for the angle \( \theta \) such that:
\[
\sin(\theta) = -\frac{1}{2}
\]
We know that \( \sin(\theta) = -\frac{1}{2} \) occurs at:
\[
\theta = -\frac{\pi}{6} \quad \text{(in Quadrant IV)}
\]
So, the exact value of \( \sin^{-1}\left( -\frac{1}{2} \right) \) is:
\[
-\frac{\pi}{6} \, \text{radians}
\]

### (b) \( \cos^{-1}\left( \frac{1}{2} \right) \)

This is asking for the angle \( \theta \) such that:
\[
\cos(\theta) = \frac{1}{2}
\]
We know that \( \cos(\theta) = \frac{1}{2} \) occurs at:
\[
\theta = \frac{\pi}{3} \quad \text{(in Quadrant I)}
\]
So, the exact value of \( \cos^{-1}\left( \frac{1}{2} \right) \) is:
\[
\frac{\pi}{3} \, \text{radians}
\]

### (c) \( \tan^{-1}\left( \frac{\sqrt{3}}{3} \right) \)

This is asking for the angle \( \theta \) such that:
\[
\tan(\theta) = \frac{\sqrt{3}}{3}
\]
We know that \( \tan(\theta) = \frac{\sqrt{3}}{3} \) occurs at:
\[
\theta = \frac{\pi}{6} \quad \text{(in Quadrant I)}
\]
So, the exact value of \( \tan^{-1}\left( \frac{\sqrt{3}}{3} \right) \) is:
\[
\frac{\pi}{6} \, \text{radians}
\]

### Summary:
- (a) \( \sin^{-1}\left( -\frac{1}{2} \right) = -\frac{\pi}{6} \)
- (b) \( \cos^{-1}\left( \frac{1}{2} \right) = \frac{\pi}{3} \)
- (c) \( \tan^{-1}\left( \frac{\sqrt{3}}{3} \right) = \frac{\pi}{6} \)

Would you like any further details or clarifications?

Here are five related questions you can consider:
1. What is the range of the inverse sine function \( \sin^{-1}(x) \)?
2. How does the unit circle help in finding inverse trigonometric values?
3. What are the principal values for \( \tan^{-1}(x) \)?
4. How can you determine when inverse trigonometric values are undefined?
5. What are the differences in behavior between \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \)?

**Tip**: Remember that inverse trigonometric functions return principal values, typically within specific ranges, depending on the function!

Find the exact value of each expression, if it is defined. Express your answer in radians.

Learn how to find the exact values of inverse sine, cosine, and tangent expressions in radians. This detailed explanation includes examples like sin^(-1)(-1/2), cos^(-1)(1/2), and tan^(-1)(√3/3), making it suitable for high school trigonometry students.

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