Question : If $\theta$ is an acute angle and $\sin \theta=\frac{21}{25}$, then what is the value of $\tan \theta$?
Option 1: $\frac{2 \sqrt{46}}{21}$
Option 2: $\frac{25}{2 \sqrt{46}}$
Option 3: $\frac{21}{2 \sqrt{46}}$
Option 4: $\frac{2 \sqrt{46}}{25}$
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Correct Answer: $\frac{21}{2 \sqrt{46}}$
Solution :
Given, $\theta < 90°$ and $\sin \theta=\frac{21}{25}$
We know, $\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$
⇒ Perpendicular = 21 and Hypotenuse = 25
Using Pythagoras theorem,
Hypotenuse
2
= Perpendicular
2
+ Base
2
⇒ 25
2
= 21
2
+ Base
2
⇒ Base = $\sqrt{25^2-21^2}$
⇒ Base = $\sqrt{625-441}$
= $\sqrt{184}$
= $2\sqrt{46}$
Now, $\tan \theta=\frac{\text{Perpendicular}}{\text{Base}}=\frac{21}{2\sqrt{46}}$
Hence, the correct answer is $\frac{21}{2\sqrt{46}}$.
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