Question : If $\sin \theta=\frac{8}{17}$, then find the value of $\tan \theta$.
Option 1: $\frac{15}{17}$
Option 2: $\frac{8}{15}$
Option 3: $\frac{15}{8}$
Option 4: $\frac{17}{15}$
Correct Answer: $\frac{8}{15}$
Solution :
Given: $\sin \theta=\frac{8}{17}=\frac{\text{perpendicular}}{\text{hypotenuse}}$
Let $\text{perpendicular}=8k$ and $\text{hypotenuse}=17k$, where $k$ is a non zero constant.
Using Pythagoras theorem,
$\small\text{Hypotenuse}^2=\text{Perpendicular}^2+\text{Base}^2$
⇒ $(17k)^2=(8k)^2+\text{Base}^2$
⇒ $\text{Base}^2=289k^2-64k^2$
⇒ $\text{Base}^2=225k^2$
⇒ $\text{Base}=15k$
So, $\tan \theta=\frac{\text{Perpendicular}}{\text{Base}}$
⇒ $\tan\theta = \frac{8k}{15k}=\frac{8}{15}$
Hence, the correct answer is $\frac{8}{15}$.
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