`DeltaXYZ` is right angled at Y.If tan X `=(15)/(8)`,then what is the value of sinZ?

Options

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

Answer

To solve the problem, we will follow these steps:

1. **Understanding the Triangle**: We have a right-angled triangle \( \Delta XYZ \) with the right angle at \( Y \). We are given that \( \tan X = \frac{15}{8} \).

2. **Identifying the Sides**: In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, for angle \( X \):
\[ \tan X = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{15}{8} \]
This means the side opposite to angle \( X \) (which is side \( YZ \)) is 15, and the side adjacent to angle \( X \) (which is side \( XY \)) is 8.

3. **Finding the Hypotenuse**: We can use the Pythagorean theorem to find the hypotenuse \( XZ \):
\[ XZ^2 = XY^2 + YZ^2 \]
Substituting the values:
\[ XZ^2 = 8^2 + 15^2 = 64 + 225 = 289 \]
Taking the square root:
\[ XZ = \sqrt{289} = 17 \]

4. **Finding \( \sin Z \)**: The sine of angle \( Z \) is given by the ratio of the opposite side to the hypotenuse. The side opposite to angle \( Z \) is \( XY \) (which is 8), and the hypotenuse is \( XZ \) (which is 17):
\[ \sin Z = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{XY}{XZ} = \frac{8}{17} \]

5. **Conclusion**: Therefore, the value of \( \sin Z \) is \( \frac{8}{17} \).

### Final Answer:
\[ \sin Z = \frac{8}{17} \]