The difference between the two acute angles of a right-angled triangle is \( \frac{2\pi}{5} \) radians. Express the angles in degrees.
Solution:
In a right-angled triangle, the sum of the two acute angles is \(90^\circ\).
Let the acute angles be \(A\) and \(B\).
\[ A + B = 90^\circ \]
Given,
\[ A – B = \frac{2\pi}{5} \]
Convert radians into degrees:
\[ \frac{2\pi}{5} \times \frac{180^\circ}{\pi} = 72^\circ \]
So,
\[ A – B = 72^\circ \]
Adding the equations:
\[ A + B = 90^\circ \]
\[ A – B = 72^\circ \]
\[ 2A = 162^\circ \]
\[ A = 81^\circ \]
Now,
\[ B = 90^\circ – 81^\circ = 9^\circ \]
Therefore, the angles are:
\[ 81^\circ \text{ and } 9^\circ \]