If `sin alpha = 12//13 , ( 0 lt alpha lt pi //2)` and `cos beta = - ( 3)/( 5) ( pi lt beta lt ( 3)/( 2) pi )`, the value of `sin ( alpha + beta )` is

Options

A. `- 56//65`
B. `16//65`
C. `56//65`
D. `- 16//65`

Answer

To find the value of \( \sin(\alpha + \beta) \) given \( \sin \alpha = \frac{12}{13} \) and \( \cos \beta = -\frac{3}{5} \), we can follow these steps:

### Step 1: Find \( \cos \alpha \)
Since \( \sin^2 \alpha + \cos^2 \alpha = 1 \):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]
\[ \cos^2 \alpha = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \]
\[ \cos \alpha = \sqrt{\frac{25}{169}} = \frac{5}{13} \]
(Hint: Use the Pythagorean identity to find the cosine value.)

### Step 2: Find \( \sin \beta \)
Since \( \sin^2 \beta + \cos^2 \beta = 1 \):
\[ \sin^2 \beta = 1 - \cos^2 \beta \]
\[ \sin^2 \beta = 1 - \left(-\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ \sin \beta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \]
(Note: Since \( \beta \) is in the third quadrant, \( \sin \beta \) is negative.)
(Hint: Again, use the Pythagorean identity to find the sine value.)

### Step 3: Use the sine addition formula
The formula for \( \sin(\alpha + \beta) \) is:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Substituting the values we found:
\[ \sin(\alpha + \beta) = \left(\frac{12}{13}\right)\left(-\frac{3}{5}\right) + \left(\frac{5}{13}\right)\left(-\frac{4}{5}\right) \]
(Hint: Apply the sine addition formula correctly.)

### Step 4: Calculate each term
Calculating the first term:
\[ \sin \alpha \cos \beta = \frac{12 \times -3}{13 \times 5} = -\frac{36}{65} \]
Calculating the second term:
\[ \cos \alpha \sin \beta = \frac{5 \times -4}{13 \times 5} = -\frac{20}{65} \]
(Hint: Ensure to multiply and simplify correctly.)

### Step 5: Combine the results
Now, combine the results from Step 4:
\[ \sin(\alpha + \beta) = -\frac{36}{65} - \frac{20}{65} = -\frac{56}{65} \]
(Hint: Combine fractions by adding the numerators.)

### Final Answer
Thus, the value of \( \sin(\alpha + \beta) \) is:
\[ \sin(\alpha + \beta) = -\frac{56}{65} \]