If `sin theta = (12)/(13) (0 lt theta lt (pi)/(2)) and cos phi = (-3)/(5) (pi lt phi lt (3pi)/(2))` then `65 sin (theta + phi) + 60 ` equals to `"_______"`

Answer

To solve the problem, we need to find the value of \( 65 \sin(\theta + \phi) + 60 \) given that \( \sin \theta = \frac{12}{13} \) and \( \cos \phi = -\frac{3}{5} \).

### Step-by-Step Solution:

1. **Identify the values of \( \sin \theta \) and \( \cos \phi \)**:
- We have \( \sin \theta = \frac{12}{13} \) (where \( 0 < \theta < \frac{\pi}{2} \)).
- We have \( \cos \phi = -\frac{3}{5} \) (where \( \pi < \phi < \frac{3\pi}{2} \)).

2. **Find \( \cos \theta \)**:
- Using the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
- Substitute \( \sin \theta \):
\[ \cos^2 \theta + \left(\frac{12}{13}\right)^2 = 1 \]
\[ \cos^2 \theta + \frac{144}{169} = 1 \]
\[ \cos^2 \theta = 1 - \frac{144}{169} = \frac{25}{169} \]
- Therefore,
\[ \cos \theta = \sqrt{\frac{25}{169}} = \frac{5}{13} \]
- Since \( \theta \) is in the first quadrant, \( \cos \theta \) is positive.

3. **Find \( \sin \phi \)**:
- Again using the Pythagorean identity:
\[ \sin^2 \phi + \cos^2 \phi = 1 \]
- Substitute \( \cos \phi \):
\[ \sin^2 \phi + \left(-\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 \phi + \frac{9}{25} = 1 \]
\[ \sin^2 \phi = 1 - \frac{9}{25} = \frac{16}{25} \]
- Therefore,
\[ \sin \phi = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \]
- Since \( \phi \) is in the third quadrant, \( \sin \phi \) is negative.

4. **Calculate \( \sin(\theta + \phi) \)** using the sine addition formula:
\[ \sin(\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi \]
- Substitute the known values:
\[ \sin(\theta + \phi) = \left(\frac{12}{13}\right)\left(-\frac{3}{5}\right) + \left(\frac{5}{13}\right)\left(-\frac{4}{5}\right) \]
- Calculate each term:
\[ = -\frac{36}{65} - \frac{20}{65} = -\frac{56}{65} \]

5. **Calculate \( 65 \sin(\theta + \phi) + 60 \)**:
\[ 65 \sin(\theta + \phi) + 60 = 65 \left(-\frac{56}{65}\right) + 60 \]
- Simplify:
\[ = -56 + 60 = 4 \]

### Final Answer:
\[ 65 \sin(\theta + \phi) + 60 = 4 \]