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If $ \tan \theta=\frac{24}{7} $, find that $ \sin \theta+\cos \theta $
Given:
$tan\ \theta = \frac{24}{7}$.
To do:
We have to find the value of \( \sin \theta+\cos \theta \).
Solution:
Let, in a triangle $ABC$ right-angled at $B$ and $\ tan\ \theta = tan\ A=\frac{24}{7}$.
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,
$sin\ \theta=\frac{Opposite}{Hypotenuse}=\frac{BC}{AC}$
$cos\ \theta=\frac{Adjacent}{Hypotenuse}=\frac{AB}{AC}$
$tan\ \theta=\frac{Opposite}{Adjacent}=\frac{BC}{AB}$
Here,
$AC^2=AB^2+BC^2$
$\Rightarrow AC^2=(7)^2+(24)^2$
$\Rightarrow AC^2=49+576$
$\Rightarrow AC=\sqrt{625}=25$
Therefore,
$sin\ \theta=\frac{BC}{AC}=\frac{24}{25}$
$cos\ \theta=\frac{AB}{AC}=\frac{7}{25}$
This implies,$sin \theta+\cos \theta=\frac{24}{25}+\frac{7}{25}$
$=\frac{31}{25}$
The value of \( \sin \theta+\cos \theta \) is \( \frac{31}{25} \).