The angles of a quadrilateral are in A.P. and the greatest angle is \(120^\circ\). Express the angles in radians.
Solution:
Let the four angles in A.P. be:
\[ a-3d,\ a-d,\ a+d,\ a+3d \]
Given greatest angle:
\[ a+3d=120^\circ \]
Sum of angles of a quadrilateral:
\[ 360^\circ \]
So,
\[ (a-3d)+(a-d)+(a+d)+(a+3d)=360^\circ \]
\[ 4a=360^\circ \]
\[ a=90^\circ \]
Now,
\[ 90^\circ+3d=120^\circ \]
\[ 3d=30^\circ \]
\[ d=10^\circ \]
Therefore, the angles are:
\[ 60^\circ,\ 80^\circ,\ 100^\circ,\ 120^\circ \]
Convert each angle into radians:
\[ 60^\circ=\frac{\pi}{3} \]
\[ 80^\circ=\frac{4\pi}{9} \]
\[ 100^\circ=\frac{5\pi}{9} \]
\[ 120^\circ=\frac{2\pi}{3} \]
Hence, the angles in radians are:
\[ \frac{\pi}{3},\ \frac{4\pi}{9},\ \frac{5\pi}{9},\ \frac{2\pi}{3} \]