If sin A = 3/5 and cos B = 12/13 , then find the value of (tan A + tan B)

Question

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

Sapling Academy · Accepted Answer

We To find the value of tan A + tan B, we need to find the value of tan A and tan B

Step 1: We are given sin A = $\frac{3}{5}$

We know that sin2 A + cos2 A = 1

∴ $(\frac{3}{5})^2$ + cos2 A = 1

∴ cos2 A = 1 - $\frac{9}{25} = \frac{16}{25}$

∴ cos A = $\sqrt{\frac{16}{25}} = \frac{4}{5}$

Now, tan A = $\frac{\sin A}{\cos A}$

By substituting value of sin A and cos A in above equation, we get:

tan A = $\frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$

Step 2: Next, we have cos B = $\frac{12}{13}$

∵ sin2 B + cos2 B = 1

∴ sin2 B +  $(\frac{12}{13})^2$ = 1

∴ sin2 B = 1 - $\frac{144}{169} = \frac{25}{169}$

∴ sin B = $\sqrt{\frac{25}{169}} = \frac{5}{13}$

Now, tan B = $\frac{\sin B}{\cos B}$

By substituting value of sin B and cos B in above equation, we get:

tan B = $\frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}$

Step 3: Let's find the value of (tan A + tan B) by substituting the values from step 1 and step 2:

(tan A + tan B) = $\frac{3}{4} + \frac{5}{12} = \frac{9 + 5}{12}$

=$ \frac{14}{12} = \frac{7}{6} = 1 \frac{1}{6}$

Therefore, the value of (tan A + tan B) is 1 $\frac{1}{6}$

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