Question : If $A+B=90^{\circ}$ and $\sin A=\frac{3}{5}$, then the value of $\tan B$ is:
Option 1: $\frac{4}{3}$
Option 2: $\frac{5}{4}$
Option 3: $\frac{3}{4}$
Option 4: $\frac{5}{3}$
Correct Answer: $\frac{4}{3}$
Solution :
Given: $A+B=90^{\circ}$ and $\sin A=\frac{3}{5}$
$\sin A=\frac{3}{5}=\frac{\text{Perpendicular}}{Hypotenuse}$
Let, Perpendicular = $3k$ and Hypotenuse = $5k$ --------------[where, $k$ is constant]
⇒ Base = $\sqrt{(5k)^{2}–(3k)^{2}}$
⇒ Base = $4k$
$\tan A=\frac{\text{Perpendicular}}{Base}$
⇒ $\tan A=\frac{3k}{4k}$
⇒ $\tan (90°–B)=\frac{3}{4}$
⇒ $\cot B=\frac{3}{4}$
⇒ $\tan B=\frac{4}{3}$
Hence, the correct answer is $\frac{4}{3}$.
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