Question

If $\sin A = \frac{3}{5}$ and $\cos B = \frac{12}{13}$, then find the value of $(\tan A + \tan B)$.

Answer 1

Step 1: Given Information:
- \( \sin A = \frac{3}{5} \)
- \( \cos B = \frac{12}{13} \)
Find: \( \tan A + \tan B \)

Step 2: Calculate \( \tan A \):
Use \( \tan A = \frac{\sin A}{\cos A} \). First, find \( \cos A \) using \( \sin^2 A + \cos^2 A = 1 \):
\[\left( \frac{3}{5} \right)^2 + \cos^2 A = 1 \Rightarrow \frac{9}{25} + \cos^2 A = 1 \Rightarrow \cos^2 A = \frac{16}{25}\]So, \( \cos A = \frac{4}{5} \).
Then, \( \tan A = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \).

Step 3: Calculate \( \tan B \):
Use \( \tan B = \frac{\sin B}{\cos B} \). First, find \( \sin B \) using \( \sin^2 B + \cos^2 B = 1 \):
\[\sin^2 B + \left( \frac{12}{13} \right)^2 = 1 \Rightarrow \sin^2 B + \frac{144}{169} = 1 \Rightarrow \sin^2 B = \frac{25}{169}\]So, \( \sin B = \frac{5}{13} \).
Then, \( \tan B = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} \).

Step 4: Sum the Tangents:
Add \( \tan A \) and \( \tan B \):
\[\tan A + \tan B = \frac{3}{4} + \frac{5}{12}\]Find a common denominator (12):
\[\frac{9}{12} + \frac{5}{12} = \frac{14}{12} = \frac{7}{6}\]

Result:
The value of \( \tan A + \tan B \) is \( \frac{7}{6} \).