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To find the values of cosec(α), sec(α), and cot(α) for the given right triangle, we can proceed by using trigonometric definitions and the triangle’s side lengths.

### Triangle Analysis:
In this triangle:
- Opposite side to angle \( \alpha \): \( 15 \, \text{cm} \)
- Adjacent side to angle \( \alpha \): \( 20 \, \text{cm} \)
- Hypotenuse: \( 32 \, \text{cm} \)

### Step 1: Trigonometric Definitions
1. **cosec(α)** = \( \frac{\text{hypotenuse}}{\text{opposite}} \)
2. **sec(α)** = \( \frac{\text{hypotenuse}}{\text{adjacent}} \)
3. **cot(α)** = \( \frac{\text{adjacent}}{\text{opposite}} \)

### Step 2: Calculations
1. **cosec(α)** = \( \frac{32}{15} \)
2. **sec(α)** = \( \frac{32}{20} = \frac{8}{5} \)
3. **cot(α)** = \( \frac{20}{15} = \frac{4}{3} \)

### Answer Analysis
Now we compare these results with the options:

- **Correct answer**: **Option (a)** 
  - \( \text{cosec} \, \alpha = \frac{32}{15} \)
  - \( \text{sec} \, \alpha = \frac{32}{20} \)
  - \( \text{cot} \, \alpha = \frac{20}{15} \)

### Final Answer
**Option (a)** is correct.

Would you like a more detailed breakdown of these calculations, or do you have other questions?

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#### Further Related Questions:
1. What is the value of sin(α) and cos(α) for this triangle?
2. How do we calculate tan(α) and its relationship to cot(α)?
3. How does one find the hypotenuse in a right triangle if only two sides are known?
4. What are some real-world applications of trigonometric ratios in right triangles?
5. How does changing one side's length affect trigonometric ratios?

#### Tip:
For right triangles, remembering the SOH-CAH-TOA mnemonic can help in recalling basic trigonometric definitions for sine, cosine, and tangent.

What are the cosec, sec, and cot for the α angle from the triangle above?

Discover how to find the values of cosec, sec, and cot for a given angle in a right triangle with side lengths 15 cm, 20 cm, and 32 cm. This problem covers trigonometric concepts such as cosecant, secant, and cotangent, with detailed calculations suitable for Grades 9-10.

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