`DeltaABC` is right angled B .If cosA `=(8)/(17)` , what is the value of cot C ?

Options

A. a.`(15)/(8)`
B. b.`(15)/(17)`
C. c.`(8)/(17)`
D. d.`(17)/(15)`

Answer

To solve the problem, we need to find the value of cot C in triangle ABC, which is right-angled at B, given that cos A = 8/17.

### Step-by-step Solution:

1. **Understanding the Triangle**:
Since triangle ABC is right-angled at B, we can label the sides as follows:
- AB = adjacent side to angle A
- BC = opposite side to angle A
- AC = hypotenuse

2. **Using the Cosine Definition**:
The cosine of angle A is defined as the ratio of the length of the adjacent side to the hypotenuse:
\[ \cos A = \frac{AB}{AC} \]
Given that \(\cos A = \frac{8}{17}\), we can set:
- AB = 8 (adjacent side)
- AC = 17 (hypotenuse)

3. **Finding the Length of BC**:
We can use the Pythagorean theorem to find the length of side BC (opposite side):
\[ AC^2 = AB^2 + BC^2 \]
Plugging in the known values:
\[ 17^2 = 8^2 + BC^2 \]
\[ 289 = 64 + BC^2 \]
\[ BC^2 = 289 - 64 = 225 \]
\[ BC = \sqrt{225} = 15 \]

4. **Finding cot C**:
The cotangent of angle C is defined as the ratio of the length of the adjacent side (AB) to the opposite side (BC):
\[ \cot C = \frac{AB}{BC} \]
Substituting the values we found:
\[ \cot C = \frac{8}{15} \]

5. **Final Answer**:
Thus, the value of cot C is:
\[ \cot C = \frac{15}{8} \]