`Delta ABC` is right angled at B. If cosA = 8/17, then what is the value of cotC ?

Options

A. `15//8`
B. `15//17`
C. `8//17`
D. `17//15`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given values
We have a right-angled triangle ABC with the right angle at B. We are given that \( \cos A = \frac{8}{17} \).

### Step 2: Identify the sides of the triangle
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side (base) to the hypotenuse. Here, for angle A:
- The adjacent side (base) is AB.
- The hypotenuse is AC.

Thus, we can write:
\[ \cos A = \frac{AB}{AC} = \frac{8}{17} \]

From this, we can conclude:
- \( AB = 8 \)
- \( AC = 17 \)

### Step 3: Use the Pythagorean theorem to find the third side
To find the length of the side BC (the opposite side to angle A), we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 17^2 = 8^2 + BC^2 \]
Calculating the squares:
\[ 289 = 64 + BC^2 \]
Now, isolate \( BC^2 \):
\[ BC^2 = 289 - 64 = 225 \]
Taking the square root:
\[ BC = \sqrt{225} = 15 \]

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

### Final Answer
Thus, the value of \( \cot C \) is:
\[ \cot C = \frac{15}{8} \]

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