If `cosec theta = (13)/(12)` , then find the values of `cot theta and sin theta`.

Answer

To solve the problem where \( \csc \theta = \frac{13}{12} \), we need to find the values of \( \cot \theta \) and \( \sin \theta \).

### Step-by-Step Solution:

1. **Understanding Cosecant**:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Given \( \csc \theta = \frac{13}{12} \), we can find \( \sin \theta \):
\[ \sin \theta = \frac{1}{\csc \theta} = \frac{12}{13} \]

2. **Using the Pythagorean Theorem**:
We know that in a right triangle:
- Hypotenuse = \( \csc \theta \times \text{Perpendicular} \)
- Here, \( \csc \theta = \frac{13}{12} \), so we can assume:
- Hypotenuse = 13 (since \( \csc \theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} \))
- Perpendicular = 12 (as derived from \( \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \))

3. **Finding the Base**:
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Substituting the values:
\[ 13^2 = 12^2 + \text{Base}^2 \]
\[ 169 = 144 + \text{Base}^2 \]
Rearranging gives:
\[ \text{Base}^2 = 169 - 144 = 25 \]
Taking the square root:
\[ \text{Base} = 5 \]

4. **Finding Cotangent**:
\[ \cot \theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{5}{12} \]

### Final Values:
- \( \cot \theta = \frac{5}{12} \)
- \( \sin \theta = \frac{12}{13} \)