To find \(\sin \theta\) given that \(\cot \theta = \frac{1}{2}\), let's proceed step by step.

### Step 1: Find \(\tan \theta\)
We already determined that:
\[
\tan \theta = 2
\]

### Step 2: Use the identity involving \(\tan \theta\) and \(\sin \theta\)
Recall the identity:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
From \(\tan \theta = 2\), we have:
\[
2 = \frac{\sin \theta}{\cos \theta}
\]
So,
\[
\sin \theta = 2\cos \theta
\]

### Step 3: Use the Pythagorean identity
We know the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \(\sin \theta = 2\cos \theta\) into this identity:
\[
(2\cos \theta)^2 + \cos^2 \theta = 1
\]
Simplify:
\[
4\cos^2 \theta + \cos^2 \theta = 1
\]
\[
5\cos^2 \theta = 1
\]
\[
\cos^2 \theta = \frac{1}{5}
\]
\[
\cos \theta = \pm \frac{1}{\sqrt{5}}
\]

### Step 4: Determine the correct sign of \(\cos \theta\)
Since \(\theta\) is in the third quadrant (based on the given interval \( -\pi \leq \theta \leq -\frac{\pi}{2} \)), \(\cos \theta\) is negative:
\[
\cos \theta = -\frac{1}{\sqrt{5}}
\]

### Step 5: Find \(\sin \theta\)
Now, substitute \(\cos \theta = -\frac{1}{\sqrt{5}}\) into the equation \(\sin \theta = 2\cos \theta\):
\[
\sin \theta = 2 \times -\frac{1}{\sqrt{5}} = -\frac{2}{\sqrt{5}}
\]

So, the value of \(\sin \theta\) is \(-\frac{2}{\sqrt{5}}\).

Would you like to discuss this further or need more clarification?

### Related Questions:
1. How would you rationalize the expression for \(\sin \theta\)?
2. What is \(\cos \theta\) and how is it related to \(\sin \theta\) in this scenario?
3. How does the Pythagorean identity help in solving trigonometric equations?
4. How can \(\sin^2 \theta\) and \(\cos^2 \theta\) be used to solve trigonometric problems?
5. What would be the value of \(\sin 2\theta\) using the double-angle identity?

### Tip:
When working with trigonometric identities, always consider the quadrant in which the angle lies to determine the correct signs of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\). This is crucial for obtaining accurate results.

Learn how to find sin theta given that cot theta equals 1/2. This problem involves using trigonometric identities and the Pythagorean identity to determine the value of sin theta. Suitable for advanced high school students studying trigonometry.

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