https://file.aiquickdraw.com/mathbot/file/17255685953335r77o66a.jpg

In the image, we have a right triangle with the following sides:

- Opposite side to angle \(\theta\) (denoted as the vertical leg) = 5
- Adjacent side to angle \(\theta\) (the base of the triangle) = 12
- Hypotenuse = 13

### Step 1: Use the triangle to determine the six trigonometric ratios.

The six trigonometric functions for an angle \(\theta\) are:

1. **Sine**: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
2. **Cosine**: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
3. **Tangent**: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
4. **Cosecant**: \(\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{1}{\sin(\theta)}\)
5. **Secant**: \(\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{1}{\cos(\theta)}\)
6. **Cotangent**: \(\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan(\theta)}\)

### Now, calculate the values:

1. **\(\sin(\theta)\)**:
   \[
   \sin(\theta) = \frac{5}{13}
   \]

2. **\(\cos(\theta)\)**:
   \[
   \cos(\theta) = \frac{12}{13}
   \]

3. **\(\tan(\theta)\)**:
   \[
   \tan(\theta) = \frac{5}{12}
   \]

4. **\(\csc(\theta)\)**:
   \[
   \csc(\theta) = \frac{13}{5}
   \]

5. **\(\sec(\theta)\)**:
   \[
   \sec(\theta) = \frac{13}{12}
   \]

6. **\(\cot(\theta)\)**:
   \[
   \cot(\theta) = \frac{12}{5}
   \]

These are the exact values of the six trigonometric functions for the angle \(\theta\) in the given triangle.

Would you like further clarification on any part of this, or additional details?

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### Related Questions:
1. What is the Pythagorean theorem, and how does it apply to this triangle?
2. How can we verify the values of the trigonometric ratios using a calculator?
3. What is the relationship between \(\sin(\theta)\) and \(\cos(\theta)\) in terms of complementary angles?
4. Why are \(\csc(\theta)\) and \(\sec(\theta)\) reciprocals of \(\sin(\theta)\) and \(\cos(\theta)\)?
5. How would these trigonometric ratios change if the triangle's sides were scaled by a constant factor?

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**Tip**: Always check the triangle's dimensions against the Pythagorean theorem before calculating trigonometric functions to ensure accuracy.

Find the exact values of the six trigonometric ratios of the angle θ in the triangle.

Learn how to find the exact values of the six trigonometric ratios for angle θ in a right triangle with sides 5, 12, and 13. This problem covers core trigonometric concepts, including sine, cosine, and tangent, and applies the Pythagorean theorem for verification. Suitable for students in Grades 9-11.

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