Prove the following:
If $ \tan \mathrm{A}=\frac{3}{4} $, then $ \sin \mathrm{A} \cos \mathrm{A}=\frac{12}{25} $
Given:
\( \tan \mathrm{A}=\frac{3}{4} \)
To do:
We have to prove that \( \sin \mathrm{A} \cos \mathrm{A}=\frac{12}{25} \).
Solution:
Let, in a triangle $ABC$ right-angled at $B$, $\tan\ A=\frac{3}{4}$.
We know that,
In a right-angled triangle $ABC$ with right angle at $B$,
By Pythagoras theorem,
$AC^2=AB^2+BC^2$
By trigonometric ratios definitions,
$sin\ \theta=\frac{Opposite}{Hypotenuse}=\frac{BC}{AC}$
$cos\ \theta=\frac{Adjacent}{Hypotenuse}=\frac{AB}{AC}$
$sec\ \theta=\frac{Hypotenuse}{Adjacent}=\frac{AC}{AB}$
$tan\ \theta=\frac{Opposite}{Adjacent}=\frac{BC}{AB}$
Here,
$AC^2=AB^2+BC^2$
$\Rightarrow (AC)^2=(4)^2+(3)^2$
$\Rightarrow AC^2=16+9$
$\Rightarrow AC=\sqrt{25}=5$
Therefore,
$\sin\ A=\frac{BC}{AC}$
$=\frac{3}{5}$
$\cos\ A=\frac{AB}{AC}$
$=\frac{4}{5}$
Therefore,
$\sin \mathrm{A} \cos \mathrm{A}=\frac{3}{5} \times \frac{4}{5}$
$=\frac{3\times4}{5\times5}$
$=\frac{12}{25}$
Hence proved.
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