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To solve this problem, let's first analyze what we see in the image.

### Given:
\[ \theta = \frac{13\pi}{6} \]
We are required to find the exact values for the following trigonometric functions:
1. \(\sin(\theta)\)
2. \(\cos(\theta)\)
3. \(\tan(\theta)\)
4. \(\cot(\theta)\)

### Step 1: Determine the reference angle
\[ \theta = \frac{13\pi}{6} \]
First, let's simplify this angle. Since \(2\pi\) is a full circle, we can subtract multiples of \(2\pi\) from \(\theta\) to find the equivalent angle between \(0\) and \(2\pi\).

\[
\theta = \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}
\]

So, the reference angle is \(\frac{\pi}{6}\).

### Step 2: Determine the quadrant
\(\frac{13\pi}{6}\) corresponds to a full \(2\pi\) rotation plus an additional \(\frac{\pi}{6}\), which lands in the first quadrant. In the first quadrant, all trigonometric functions are positive.

### Step 3: Calculate the trigonometric functions
Using the reference angle \(\frac{\pi}{6}\):

1. **\(\sin(\theta)\):**
   \[
   \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
   \]

2. **\(\cos(\theta)\):**
   \[
   \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
   \]

3. **\(\tan(\theta)\):**
   \[
   \tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
   \]

4. **\(\cot(\theta)\):**
   \[
   \cot\left(\frac{\pi}{6}\right) = \frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}
   \]

### Final Answers:
\[
\sin\left(\frac{13\pi}{6}\right) = \frac{1}{2}
\]
\[
\cos\left(\frac{13\pi}{6}\right) = \frac{\sqrt{3}}{2}
\]
\[
\tan\left(\frac{13\pi}{6}\right) = \frac{\sqrt{3}}{3}
\]
\[
\cot\left(\frac{13\pi}{6}\right) = \sqrt{3}
\]

Would you like further details on any part of this solution? 

### Related Questions:
1. How do you find the reference angle for angles in different quadrants?
2. What are the properties of trigonometric functions in each quadrant?
3. How can you use the unit circle to determine the values of trigonometric functions?
4. Why is it important to know the quadrant in which an angle lies?
5. What are the exact values of trigonometric functions for other key angles like \(\frac{\pi}{4}\) or \(\frac{\pi}{3}\)?

### Tip:
Always simplify the angle to a value between \(0\) and \(2\pi\) when working with trigonometric functions, as this helps in easily identifying the reference angle and the corresponding quadrant.

Learn how to solve trigonometric functions for the angle 13π/6 with a detailed step-by-step solution. This problem covers key trigonometric concepts, including reference angles and trigonometric identities. Suitable for high school students studying trigonometry.

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