To find \(\sin a\) given that \(\cos 2a = -\frac{1}{2}\), let's follow the steps below.

### Step 1: Use the double-angle identity for cosine
The double-angle identity for cosine is:
\[
\cos 2a = 2\cos^2 a - 1
\]
Given \(\cos 2a = -\frac{1}{2}\), we can set up the equation:
\[
2\cos^2 a - 1 = -\frac{1}{2}
\]

### Step 2: Solve for \(\cos^2 a\)
Add 1 to both sides of the equation:
\[
2\cos^2 a = 1 - \frac{1}{2} = \frac{1}{2}
\]
Then divide by 2:
\[
\cos^2 a = \frac{1}{4}
\]

### Step 3: Find \(\cos a\)
Taking the square root of both sides, we get:
\[
\cos a = \pm \frac{1}{2}
\]

### Step 4: Use the Pythagorean identity to find \(\sin a\)
The Pythagorean identity states:
\[
\sin^2 a + \cos^2 a = 1
\]
Substitute \(\cos^2 a = \frac{1}{4}\) into the identity:
\[
\sin^2 a + \frac{1}{4} = 1
\]
Subtract \(\frac{1}{4}\) from both sides:
\[
\sin^2 a = \frac{3}{4}
\]
Taking the square root of both sides:
\[
\sin a = \pm \frac{\sqrt{3}}{2}
\]

### Step 5: Determine the sign of \(\sin a\)
The sign of \(\sin a\) depends on the quadrant where angle \(a\) lies. 

Since \(\cos 2a = -\frac{1}{2}\), \(2a\) must lie in either the second or third quadrant, where cosine is negative. Depending on which quadrant \(2a\) lies in, the angle \(a\) could be in either the first or second quadrant. In the first quadrant, \(\sin a\) is positive, and in the second quadrant, \(\sin a\) is positive as well.

Thus, \(\sin a = \frac{\sqrt{3}}{2}\).

### Final Answer:
\[
\sin a = \frac{\sqrt{3}}{2}
\]

Would you like more details or have any other questions?

Here are five related questions:

1. What is \(\cos a\) if \(\cos 2a = -\frac{1}{2}\)?
2. How would the answer change if \(\cos 2a = \frac{1}{2}\)?
3. What is the value of \(a\) if \(\cos 2a = -\frac{1}{2}\) and \(a\) is in the first quadrant?
4. How does the value of \(\sin a\) change if \(a\) is in the third quadrant?
5. What is the double-angle formula for sine and how can it be used in similar problems?

**Tip:** Always check the quadrant to determine the correct sign for trigonometric functions.

Learn how to find sin(a) when cos(2a) = -1/2 using trigonometric identities, including the double-angle formula and the Pythagorean identity. This step-by-step solution is suitable for high school students in Grades 10-12.

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