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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