Question : If in a right-angled $\triangle P Q R$, $\tan Q=\frac{5}{12}$, then what is the value of $\cos Q$?
Option 1: $\frac{12}{5}$
Option 2: $\frac{13}{5}$
Option 3: $\frac{12}{13}$
Option 4: $\frac{5}{13}$
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Correct Answer: $\frac{12}{13}$
Solution :
Given, $\tan Q = \frac{5}{12}$
We know, $\tan\theta = \frac{\text{Perpendicular}}{\text{Base}}$
Applying Pythagoras theorem,
$\small\text{Hypotenuse}^2 = \text{Perpendicular}^2+\text{Base}^2$
⇒ $\text{H}^2=5^2+12^2$
⇒ $\text{H}^2 = 25 + 144$
⇒ $\text{H}=\sqrt{169}$
⇒ $\text{H} = 13\ \text{units}$
Now, $\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}$
⇒ $\cos Q = \frac{12}{13}$
Hence, the correct answer is $\frac{12}{13}$.
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