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### Problem 1: Evaluate the expression \(\frac{\sqrt{2}}{2}\) and express it in radians.
This appears to be a trigonometric value in radians.

Given:
\[
\frac{\sqrt{2}}{2}
\]

This value corresponds to the cosine or sine of \( \frac{\pi}{4} \) radians.

So, \( \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) or \( \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

### Problem 2: Solve the equation \( 2 \cos(\theta) = 1 \) for \( \theta \).
Given:
\[
2 \cos(\theta) = 1
\]
Divide both sides by 2:
\[
\cos(\theta) = \frac{1}{2}
\]
This corresponds to angles \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) radians, within the interval \( [0, 2\pi) \).

### Problem 3: Solve the equation \( \sin(\theta) = -\frac{1}{2} \) for \( \theta \).
Given:
\[
\sin(\theta) = -\frac{1}{2}
\]
This corresponds to angles \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \) radians, within the interval \( [0, 2\pi) \).

### Problem 4: Solve the inequality \( \cos(x) > \frac{1}{2} \) in radians.
Given:
\[
\cos(x) > \frac{1}{2}
\]
The solution corresponds to the intervals:
\[
x \in \left(-\frac{\pi}{3}, \frac{\pi}{3}\right) \cup \left(\frac{5\pi}{3}, 2\pi\right)
\]

These intervals represent the regions where the cosine function is greater than \( \frac{1}{2} \).

### Problem 5: Evaluate the expression \( \cos^{-1}\left(\frac{1}{2}\right) \).
Given:
\[
\cos^{-1}\left(\frac{1}{2}\right)
\]
This corresponds to:
\[
\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}
\]

Would you like a detailed explanation for any of these solutions? Here are some related questions that could help reinforce your understanding:

1. What are the key properties of the cosine and sine functions that help solve trigonometric equations?
2. How can you determine the general solution for a trigonometric equation like \( \sin(\theta) = \frac{1}{2} \)?
3. Why are there two solutions for each equation within the interval \( [0, 2\pi) \)?
4. How can you express trigonometric inequalities using intervals on the unit circle?
5. What are the implications of inverse trigonometric functions on the domain and range of solutions?

**Tip:** Always check your solutions within the original equation or inequality to ensure they satisfy all conditions.

Learn how to solve trigonometric equations involving cosine and sine functions. This problem set covers key concepts in trigonometry, including the evaluation of trigonometric expressions, solving trigonometric equations, and understanding trigonometric inequalities. Suitable for students in Grades 10-12.

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