`DeltaXYZ` is right angled at Y. If `sinX=4//5`, then what is the value of `cosZ` ?

Options

A. `3//4`
B. `5//3`
C. `4//5`
D. `4//3`

Answer

To solve the problem, we need to find the value of \( \cos Z \) given that \( \sin X = \frac{4}{5} \) in the right-angled triangle \( \Delta XYZ \) which is right-angled at \( Y \).

### Step-by-step Solution:

1. **Understanding the Triangle**:
- We have a right-angled triangle \( \Delta XYZ \) with the right angle at \( Y \).
- The angles are \( X \), \( Y \), and \( Z \).

2. **Using the Sine Function**:
- We know that \( \sin X = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
- Here, the side opposite to angle \( X \) is \( YZ \) and the hypotenuse is \( XZ \).
- Therefore, we can write:
\[ \sin X = \frac{YZ}{XZ} = \frac{4}{5} \]
- This implies that \( YZ = 4k \) and \( XZ = 5k \) for some positive constant \( k \).

3. **Finding the Third Side**:
- To find the length of \( YX \), we can use the Pythagorean theorem:
\[ XZ^2 = YZ^2 + YX^2 \]
- Substituting the known values:
\[ (5k)^2 = (4k)^2 + YX^2 \]
\[ 25k^2 = 16k^2 + YX^2 \]
\[ YX^2 = 25k^2 - 16k^2 = 9k^2 \]
\[ YX = 3k \]

4. **Finding \( \cos Z \)**:
- Now, we can find \( \cos Z \). The cosine function is defined as:
\[ \cos Z = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
- In the context of angle \( Z \):
- The side adjacent to angle \( Z \) is \( YX \) and the hypotenuse is \( XZ \).
- Therefore:
\[ \cos Z = \frac{YX}{XZ} = \frac{3k}{5k} = \frac{3}{5} \]

5. **Final Answer**:
- Thus, the value of \( \cos Z \) is:
\[ \cos Z = \frac{3}{5} \]