Prove that: If tan A =frac{3}{4}, then sin A cos A =frac{12}{25}

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8. Introduction to Trigonometry

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Prove that:

If $\tan A =\frac{3}{4},$ then $\sin A \cos A =\frac{12}{25}$

Option A

Option B

Option C

Option D

Given, $\quad \tan A=\frac{3}{4}=\frac{P}{B}=\frac{\text { Perpendicular }}{\text { Base }}$

Let $P=3 k$ and $B=4 k$

By Pythagoras theorem,

$H^{2}=P^{2}+B^{2}=(3 k)^{2}+(4 k)^{2}$

$=9 k^{2}+16 k^{2}=25 k^{2}$

$\Rightarrow \quad H=5 k$ [since, side cannot be negative]

$\sin A=\frac{P}{H}=\frac{3 k}{5 k}=\frac{3}{5}$ and $\cos A=\frac{B}{H}=\frac{4 k}{5 k}=\frac{4}{5}$

Now, $\sin A \cos A=\frac{3}{5} \cdot \frac{4}{5}=\frac{12}{25}$ Hence proved.

Standard 10

Mathematics

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