Question : If $3 \tan \theta=4$, then what is the value of $7 \sin \theta-2 \cos \theta$?
Option 1: $\frac{12}{5}$
Option 2: $4$
Option 3: $\frac{22}{5}$
Option 4: $\frac{17}{5}$
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Correct Answer: $\frac{22}{5}$
Solution :
Given,
$3 \tan \theta=4$
⇒ $\tan\theta=\frac43=\frac{\text{Perpendicular}}{\text{Base}}$
Using the Pythagoras theorem,
$\small\text{Hypotenuse}^2=\text{Perpendicular}^2+\text{Base}^2$
⇒ $h^2=4^2+3^2$
⇒ $h^2=16+9$
⇒ $h^2=25$
⇒ $h=5$
We know, $\sin\theta=\frac{\text{Perpendicular}}{\text{Hypotenuse}}$ and $\cos\theta=\frac{\text{Base}}{\text{Hypotenuse}}$
⇒ $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac35$
So, $7 \sin \theta-2 \cos \theta=7\times\frac{4}{5}-2\times\frac{3}{5}=\frac{28}{5}-\frac{6}{5}=\frac{22}{5}$
Hence, the correct answer is $\frac{22}{5}$.
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