If `0^(@)lt theta lt 90^(@)` and `cos theta=(4)/(5)` find the values of
`tan(360^(@)-theta)`

Answer

To solve the problem, we need to find the value of \( \tan(360^\circ - \theta) \) given that \( 0^\circ < \theta < 90^\circ \) and \( \cos \theta = \frac{4}{5} \).

### Step-by-Step Solution:

1. **Identify the given values**:
- We know that \( \theta \) is in the first quadrant (since \( 0^\circ < \theta < 90^\circ \)).
- We are given \( \cos \theta = \frac{4}{5} \).

2. **Find \( \sin \theta \)**:
- Use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
- Substitute \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
- Calculate \( \left(\frac{4}{5}\right)^2 = \frac{16}{25} \):
\[ \sin^2 \theta + \frac{16}{25} = 1 \]
- Rearranging gives:
\[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
- Taking the square root (since \( \theta \) is in the first quadrant, we take the positive root):
\[ \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

3. **Calculate \( \tan \theta \)**:
- We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
- Substitute the values:
\[ \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

4. **Find \( \tan(360^\circ - \theta) \)**:
- Using the identity:
\[ \tan(360^\circ - \theta) = -\tan \theta \]
- Substitute \( \tan \theta \):
\[ \tan(360^\circ - \theta) = -\frac{3}{4} \]

5. **Final Result**:
- Therefore, the value of \( \tan(360^\circ - \theta) \) is:
\[ \tan(360^\circ - \theta) = -\frac{3}{4} \]