https://file.aiquickdraw.com/mathbot/file/17270284760464o6yhdlb.jpg

To solve the given problem, let's analyze the triangle and find the values of \(\cot \theta\), \(\csc \theta\), and \(\sin \theta\) using trigonometric identities.

### Step 1: Identify the sides of the triangle
The right triangle shows:
- Opposite side to \(\theta\): \(15\)
- Adjacent side to \(\theta\): Unknown for now
- Hypotenuse: \(17\)

We can find the adjacent side using the Pythagorean theorem:
\[
\text{Adjacent side} = \sqrt{17^2 - 15^2}
\]
Let's calculate it:
\[
17^2 = 289, \quad 15^2 = 225
\]
\[
289 - 225 = 64
\]
\[
\text{Adjacent side} = \sqrt{64} = 8
\]

Now, we have:
- Opposite side: \(15\)
- Adjacent side: \(8\)
- Hypotenuse: \(17\)

### Step 2: Find \(\cot \theta\), \(\csc \theta\), and \(\sin \theta\)

1. \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{15}{17}\)
2. \(\csc \theta = \frac{1}{\sin \theta} = \frac{17}{15}\)
3. \(\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{8}{15}\)

### Final answers:
- \(\cot \theta = \frac{8}{15}\)
- \(\csc \theta = \frac{17}{15}\)
- \(\sin \theta = \frac{15}{17}\)

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. How do you find the remaining trigonometric ratios (\(\cos \theta\), \(\tan \theta\), \(\sec \theta\)) for this triangle?
2. What is the value of \(\theta\) in degrees or radians?
3. How does the Pythagorean identity relate to this triangle?
4. How can you determine \(\theta\) if given only \(\cot \theta\)?
5. Why is \(\csc \theta\) the reciprocal of \(\sin \theta\)?

### Tip:
In right triangles, always start with the Pythagorean theorem to find missing sides, which helps in determining all trigonometric ratios accurately.

Find cot(θ), csc(θ), and sin(θ) where θ is the angle shown in the figure, using exact values.

This problem demonstrates how to find the exact values of cot(θ), csc(θ), and sin(θ) for a right triangle using trigonometric formulas and the Pythagorean theorem. The triangle has a hypotenuse of 17 and an opposite side of 15.

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