If `0^(@)lt theta lt 90^(@)` and `cos theta=(4)/(5)` find the values of
`sin (270^(@)-theta)`

Answer

To solve the problem, we need to find the value of \( \sin(270^\circ - \theta) \) given that \( \cos \theta = \frac{4}{5} \) and \( 0^\circ < \theta < 90^\circ \).

### Step-by-Step Solution:

1. **Understanding the Quadrant**:
Since \( \theta \) is between \( 0^\circ \) and \( 90^\circ \), it lies in the first quadrant where all trigonometric functions are positive.

2. **Using the Sine Difference Identity**:
We can use the identity for sine of a difference:
\[ \sin(270^\circ - \theta) = \sin 270^\circ \cos \theta - \cos 270^\circ \sin \theta \]

3. **Values of Sine and Cosine at Key Angles**:
We know:
- \( \sin 270^\circ = -1 \)
- \( \cos 270^\circ = 0 \)

4. **Substituting Values into the Identity**:
Substitute the known values into the sine difference identity:
\[ \sin(270^\circ - \theta) = (-1) \cdot \cos \theta - 0 \cdot \sin \theta \]
This simplifies to:
\[ \sin(270^\circ - \theta) = -\cos \theta \]

5. **Using the Given Value of Cosine**:
We know \( \cos \theta = \frac{4}{5} \). Therefore:
\[ \sin(270^\circ - \theta) = -\frac{4}{5} \]

6. **Final Answer**:
Thus, the value of \( \sin(270^\circ - \theta) \) is:
\[ \sin(270^\circ - \theta) = -\frac{4}{5} \]