`DeltaDEF` is right angled at E. If sin `D=(15)/(17)`, what is the value of cot F ?

Options

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

Answer

To solve the problem, we need to find the value of cot F in triangle DEF, which is right-angled at E, given that sin D = 15/17.

### Step-by-step Solution:

1. **Understanding the Triangle**:
- Triangle DEF is right-angled at E.
- We know that sin D = opposite/hypotenuse = EF/DF.
- Given sin D = 15/17, we can identify that EF (the side opposite angle D) = 15 and DF (the hypotenuse) = 17.

2. **Finding DE**:
- We can use the Pythagorean theorem to find the length of DE (the side adjacent to angle D).
- According to the Pythagorean theorem:
\[ DF^2 = EF^2 + DE^2 \]
- Substituting the known values:
\[ 17^2 = 15^2 + DE^2 \]
\[ 289 = 225 + DE^2 \]
- Rearranging gives:
\[ DE^2 = 289 - 225 = 64 \]
- Taking the square root:
\[ DE = \sqrt{64} = 8 \]

3. **Finding cot F**:
- Cotangent of angle F (cot F) is defined as the ratio of the adjacent side to the opposite side:
\[ \cot F = \frac{DE}{EF} \]
- We have DE = 8 and EF = 15, so:
\[ \cot F = \frac{8}{15} \]

### Final Answer:
The value of cot F is \( \frac{8}{15} \).