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Q.
In a right angled triangle, the difference between two acute angles is $\frac{\pi}{9}$ in circular measure. Express the angles in degrees.
Trigonometric Functions
Solution:
Since the triangle is right angled, therefore, the sum of the two acute angles is $90^{\circ}$. $(\because$ third angle $=90^{\circ})$
Let the two acute angles be $x$ and $y$, $x > y$
then $x + y = 90^{\circ} \quad \ldots(i)$
Also, $x-y=\frac{\pi}{9}$ radian $\,$ (given)
$\Rightarrow x-y=20^{\circ}\quad\ldots\left(ii\right)$
Adding $\left(i\right)$ and $\left(ii\right)$, we get $2x = 110^{\circ}$
$\Rightarrow x=55^{\circ}$,
Putting it in $\left(i\right)$, we get $55^{\circ}+y=90^{\circ}$
$\Rightarrow y=35^{\circ}$,
Hence, the two acute angles are $55^{\circ}, 35^{\circ}$.