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To solve for the trigonometric ratios (cotangent, secant, and cosine) given the right triangle with sides of 12, 13, and an unknown side opposite the angle \( \theta \), we first need to find the length of the missing side using the Pythagorean theorem.

### Step 1: Apply the Pythagorean Theorem
The Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]
where \( c = 13 \) (hypotenuse), \( b = 12 \) (adjacent to \( \theta \)), and \( a \) is the unknown side (opposite to \( \theta \)).

\[
a^2 + 12^2 = 13^2
\]
\[
a^2 + 144 = 169
\]
\[
a^2 = 169 - 144 = 25
\]
\[
a = \sqrt{25} = 5
\]

Now we know the side opposite \( \theta \) is 5.

### Step 2: Calculate the trigonometric ratios

1. **cotangent \( \cot(\theta) \)**: 
   \[
   \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{12}{5}
   \]

2. **secant \( \sec(\theta) \)**:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{13}{12}
   \]

3. **cosine \( \cos(\theta) \)**:
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}
   \]

### Final Answers:
- \( \cot(\theta) = \frac{12}{5} \)
- \( \sec(\theta) = \frac{13}{12} \)
- \( \cos(\theta) = \frac{12}{13} \)

Let me know if you want more details or have any questions!

Here are 5 related questions to deepen your understanding:
1. How do you find the sine and tangent of \( \theta \) in this triangle?
2. Can you use the inverse trigonometric functions to find the angle \( \theta \) itself?
3. What happens to the trigonometric ratios if the triangle were scaled up or down?
4. How does the Pythagorean identity relate to the values of sine and cosine?
5. What is the relationship between \( \cot(\theta) \) and \( \tan(\theta) \)?

**Tip:** Remember that secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent, respectively!

Find cot(θ), sec(θ), and cos(θ) for a right triangle with sides 12, 13, and an unknown side using the Pythagorean theorem.

This page explains how to find the trigonometric ratios cotangent, secant, and cosine for a right triangle with sides 12, 13, and an unknown side using the Pythagorean theorem. Suitable for high school students learning trigonometry.

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