If cos A`=(5)/(13)`, and sin `B=(4)/(5)`, find sin (A+B), where A,B,(A+B) are positive acute angles.

Answer

To find \( \sin(A + B) \) given \( \cos A = \frac{5}{13} \) and \( \sin B = \frac{4}{5} \), we can use the sine addition formula:

\[ \sin(A + B) = \sin A \cos B + \sin B \cos A \]

### Step 1: Find \( \sin A \)

We know that:

\[ \sin^2 A + \cos^2 A = 1 \]

Substituting the value of \( \cos A \):

\[ \sin^2 A + \left(\frac{5}{13}\right)^2 = 1 \]

Calculating \( \left(\frac{5}{13}\right)^2 \):

\[ \sin^2 A + \frac{25}{169} = 1 \]

Now, subtract \( \frac{25}{169} \) from both sides:

\[ \sin^2 A = 1 - \frac{25}{169} \]

Finding a common denominator:

\[ \sin^2 A = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \]

Taking the square root:

\[ \sin A = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

### Step 2: Find \( \cos B \)

Using the identity:

\[ \sin^2 B + \cos^2 B = 1 \]

Substituting the value of \( \sin B \):

\[ \left(\frac{4}{5}\right)^2 + \cos^2 B = 1 \]

Calculating \( \left(\frac{4}{5}\right)^2 \):

\[ \frac{16}{25} + \cos^2 B = 1 \]

Subtracting \( \frac{16}{25} \) from both sides:

\[ \cos^2 B = 1 - \frac{16}{25} \]

Finding a common denominator:

\[ \cos^2 B = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

Taking the square root:

\[ \cos B = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 3: Substitute into the sine addition formula

Now we have:

- \( \sin A = \frac{12}{13} \)
- \( \cos A = \frac{5}{13} \)
- \( \sin B = \frac{4}{5} \)
- \( \cos B = \frac{3}{5} \)

Substituting these values into the sine addition formula:

\[ \sin(A + B) = \sin A \cos B + \sin B \cos A \]

Calculating:

\[ \sin(A + B) = \left(\frac{12}{13} \cdot \frac{3}{5}\right) + \left(\frac{4}{5} \cdot \frac{5}{13}\right) \]

Calculating each term:

1. \( \frac{12}{13} \cdot \frac{3}{5} = \frac{36}{65} \)
2. \( \frac{4}{5} \cdot \frac{5}{13} = \frac{20}{65} \)

Adding these results:

\[ \sin(A + B) = \frac{36}{65} + \frac{20}{65} = \frac{56}{65} \]

### Final Answer

Thus, the value of \( \sin(A + B) \) is:

\[ \sin(A + B) = \frac{56}{65} \]

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