Let the degree be denoted by 'D', 'R' be the number of radians and 'G' be the number of grades in an angle $\Theta$.
We know that, $90^{0}$ = 1 right angle

⇒ $1^{0} = \frac{1}{90}$ right angle

⇒ $D^{0} = \frac{D}{90}$ right angles

⇒ $\Theta = \frac{D}{90}$ right angle -----------(i)

Again we know that $\pi$ radians = 2 right angles = $180^{0}$

⇒ $1^{c} = \frac{2}{\pi}$ right angle

⇒ R radians = $\frac{2R}{\pi}$ right angles ----------(ii)

And now, 100 grades = 1 right angle

⇒ 1 grade = = $\frac{1}{100}$ right angle

⇒ G grade = $\frac{G}{100}$ right angles

⇒ $\Theta = \frac{G}{100}$ right angles --------(iii)

Thus, from (i), (ii) and (iii), we get,

$\frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi}$

(i) The angle between two consecutive digits in a clock is $30^{0}$. In radians it is $\frac{\pi}{6}$
(ii) The hour hand rotates through an angle of $30^{0}$ in one hour which is $\left ( \frac{1}{2} \right )^{0}$ in one minute.
(iii) The minute hand rotates through an angle of $6^{0}$ in one minute.
Example :
One angle of a triangle is $\left ( \frac{2x}{3} \right )$ grades = $\left ( \frac{2x}{3} \right )^{g}$ and another is $\left ( \frac{3x}{2} \right )^{0}$ while the third is $\left ( \frac{\pi x}{75} \right )$ radians. Express all the angles in degrees.
Solution : We know that ,
$\frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi}$

⇒ $\left ( \frac{2x}{3} \right )^{g} = \left ( \frac{2}{3} x\times\frac{90}{100} \right )^{0} = \left ( \frac{3x}{5} \right )^{0}$ ----(i)

⇒ $\left ( \frac{\pi x}{75} \right )^{c} = \left ( \frac{\pi}{75} x\times\frac{180}{\pi} \right )^{c} = \left ( \frac{12x}{5} \right )^{0}$ ----(ii)

Sum of all angles in a triangle is $180^{0}$
∴ $\left ( \frac{3}{5}x \right )^{0} + \left ( \frac{3}{2}x \right )^{0} + \left ( \frac{12x}{5} \right )^{0} = 180^{0}$

⇒ x = $40^{0}$
So each angle will be
$\left ( \frac{3}{5}x \right )^{0} = \left ( \frac{3}{5}\times 40 \right )^{0} = 24^{0}$

$\left ( \frac{3}{2}x \right )^{0} = \left ( \frac{3}{2}\times 40 \right )^{0} = 60^{0}$

$\left ( \frac{12}{5}x \right )^{0} = \left ( \frac{12}{5}\times 40 \right )^{0} = 96^{0}$

So the required angles are $24^{0}, 60^{0},96^{0}$