One angle of a triangle is \( \frac{2x}{3} \) grade and another is \( \frac{3x}{2}^\circ \) while the third is \( \frac{\pi x}{75} \) radians. Express all the angles in degrees.
Solution:
We know:
\[ 1 \text{ grade} = \frac{9^\circ}{10} \]
and
\[ 1 \text{ radian} = \frac{180^\circ}{\pi} \]
First angle:
\[ \frac{2x}{3} \text{ grade} = \frac{2x}{3} \times \frac{9^\circ}{10} = \frac{3x}{5}^\circ \]
Second angle:
\[ \frac{3x}{2}^\circ \]
Third angle:
\[ \frac{\pi x}{75} \times \frac{180^\circ}{\pi} = \frac{12x}{5}^\circ \]
Sum of angles of a triangle is \(180^\circ\).
\[ \frac{3x}{5} + \frac{3x}{2} + \frac{12x}{5} = 180 \]
\[ \frac{6x + 15x + 24x}{10} = 180 \]
\[ 45x = 1800 \]
\[ x = 40 \]
Now,
First angle:
\[ \frac{3x}{5} = \frac{3 \times 40}{5} = 24^\circ \]
Second angle:
\[ \frac{3x}{2} = \frac{3 \times 40}{2} = 60^\circ \]
Third angle:
\[ \frac{12x}{5} = \frac{12 \times 40}{5} = 96^\circ \]
Therefore, the angles are:
\[ 24^\circ,\ 60^\circ,\ 96^\circ \]