`Delta DEF` is right angled at E. If secD = 17/8, then what is the value of cosF ?

Options

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

Answer

To solve the problem, we need to find the value of cosF in the right-angled triangle DEF, where the angle E is the right angle and secD is given as 17/8.

### Step-by-Step Solution:

1. **Understanding Secant**:
- The secant of an angle in a right triangle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side.
- Given secD = 17/8, we can denote the hypotenuse (DF) as 17 and the adjacent side (DE) as 8.

2. **Using Pythagorean Theorem**:
- In triangle DEF, we can apply the Pythagorean theorem:
\[ DF^2 = DE^2 + EF^2 \]
- Substituting the known values:
\[ 17^2 = 8^2 + EF^2 \]
- This simplifies to:
\[ 289 = 64 + EF^2 \]
- Rearranging gives:
\[ EF^2 = 289 - 64 = 225 \]
- Taking the square root, we find:
\[ EF = \sqrt{225} = 15 \]

3. **Finding Cosine of Angle F**:
- Now, we need to find cosF. In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse.
- For angle F, the adjacent side is DE (which is 8) and the hypotenuse is DF (which is 17):
\[ \cos F = \frac{DE}{DF} = \frac{8}{17} \]

4. **Conclusion**:
- Therefore, the value of cosF is:
\[ \cos F = \frac{8}{17} \]

### Final Answer:
\[ \cos F = \frac{8}{17} \]