Evaluate the following expressions. Your answer must be an angle −π/2≤θ≤π
 in radians.
sin−1(−12)
 

sin−1(3√2)
 

cos−1(−2√2)
 

cos−1(0)
 

Let's evaluate the given inverse trigonometric expressions, ensuring the answers are within the range \( -\frac{\pi}{2} \leq \theta \leq \pi \), where applicable.

1. **\( \sin^{-1}\left(-\frac{1}{2}\right) \):**

   The inverse sine function \( \sin^{-1}(x) \) gives the angle \( \theta \) such that \( \sin(\theta) = x \). The range for \( \theta \) is \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

   \( \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \), since \( \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \).

2. **\( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \):**

   \( \sin^{-1}(x) \) gives the angle \( \theta \) where \( \sin(\theta) = x \), with the same range of \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

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

3. **\( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) \):**

   The inverse cosine function \( \cos^{-1}(x) \) gives the angle \( \theta \) such that \( \cos(\theta) = x \), where \( 0 \leq \theta \leq \pi \).

   \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \), since \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

4. **\( \cos^{-1}(0) \):**

   \( \cos^{-1}(0) \) is the angle \( \theta \) such that \( \cos(\theta) = 0 \), within the range \( 0 \leq \theta \leq \pi \).

   \( \cos^{-1}(0) = \frac{\pi}{2} \), since \( \cos\left(\frac{\pi}{2}\right) = 0 \).

### Final Answers:
1. \( \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \)
2. \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \)
3. \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \)
4. \( \cos^{-1}(0) = \frac{\pi}{2} \)

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. Evaluate \( \sin^{-1}(1) \).
2. Evaluate \( \cos^{-1}(1) \).
3. Find the value of \( \tan^{-1}(1) \).
4. Evaluate \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \).
5. Find the value of \( \cos^{-1}(-1) \).

**Tip:** Always check the range of inverse trigonometric functions before determining the final angle.

Evaluate the following expressions. Your answer must be an angle −π/2≤θ≤π in radians. sin−1(−12), sin−1(3√2), cos−1(−2√2), cos−1(0)

Step-by-step guide to evaluating inverse trigonometric expressions such as sin−1(−1/2) and cos−1(0) within the range −π/2 ≤ θ ≤ π. Learn how to apply the inverse sine and cosine functions and understand their ranges. Suitable for high school students in Grades 10-12.

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