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### Problem 91: \( \sin^{-1}(0) \)
- The inverse sine of \( 0 \) asks for the angle whose sine value is 0. 
- \( \sin^{-1}(0) = 0 \), because the sine of \( 0 \) radians is 0.

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### Problem 94: \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \)
- The inverse cosine of \( \frac{\sqrt{3}}{2} \) asks for the angle whose cosine value is \( \frac{\sqrt{3}}{2} \).
- The angle whose cosine is \( \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{6} \) radians, or 30 degrees.
  
Thus, \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \).

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### Problem 97: \( \tan^{-1}(\sqrt{3}) \)
- The inverse tangent of \( \sqrt{3} \) asks for the angle whose tangent value is \( \sqrt{3} \).
- The angle whose tangent is \( \sqrt{3} \) is \( \frac{\pi}{3} \) radians, or 60 degrees.

Thus, \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

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### Problem 90: \( \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) \)
- The inverse tangent of \( \frac{\sqrt{3}}{3} \) asks for the angle whose tangent value is \( \frac{\sqrt{3}}{3} \).
- The angle whose tangent is \( \frac{\sqrt{3}}{3} \) is \( \frac{\pi}{6} \) radians, or 30 degrees.

Thus, \( \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6} \).

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### Problem 93: \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \)
- The inverse sine of \( \frac{\sqrt{3}}{2} \) asks for the angle whose sine value is \( \frac{\sqrt{3}}{2} \).
- The angle whose sine is \( \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \) radians, or 60 degrees.

Thus, \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \).

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Would you like any additional details or explanations on these solutions?

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Here are five related questions for practice:
1. What is \( \sin^{-1}(1) \)?
2. What is \( \cos^{-1}(-1) \)?
3. Solve \( \tan^{-1}(-1) \).
4. What is \( \sin^{-1}\left(\frac{1}{2}\right) \)?
5. Solve \( \cos^{-1}(0) \).

**Tip:** When working with inverse trigonometric functions, always consider the principal values or the specific range of angles where the function is defined.

Evaluate the following inverse trigonometric expressions without using a calculator: 
1) tan^{-1}(√3/3)
2) sin^{-1}(0)
3) cos^{-1}(√3/2)
4) tan^{-1}(√3)
5) sin^{-1}(√3/2)

Learn how to solve inverse trigonometric functions such as tan^{-1}(√3/3), sin^{-1}(0), cos^{-1}(√3/2), and more. Detailed solutions are provided with step-by-step explanations of the key trigonometric identities. Suitable for students in Grades 10-12.

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