If `cot theta = (8)/(15)` then find the values of `sin theta` and `sec theta`.

Answer

To solve the problem, we need to find the values of \( \sin \theta \) and \( \sec \theta \) given that \( \cot \theta = \frac{8}{15} \).

### Step-by-step Solution:

1. **Understanding Cotangent**:
We know that \( \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \). Given \( \cot \theta = \frac{8}{15} \), we can assign:
- Adjacent side (base) = 8
- Opposite side (perpendicular) = 15

2. **Finding the Hypotenuse**:
We can use the Pythagorean theorem to find the hypotenuse \( h \):
\[ h = \sqrt{(\text{Opposite})^2 + (\text{Adjacent})^2} = \sqrt{15^2 + 8^2} \]
Calculating this gives:
\[ h = \sqrt{225 + 64} = \sqrt{289} = 17 \]

3. **Calculating Sine**:
The sine function is defined as:
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{15}{17} \]

4. **Calculating Secant**:
The secant function is defined as:
\[ \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{17}{8} \]

### Final Values:
- \( \sin \theta = \frac{15}{17} \)
- \( \sec \theta = \frac{17}{8} \)