Radian is a measure of the trigonometric angle. The angle formed by an arc in a circle is exactly one radian and it is equal to the radius of the circle. We use radian as a unit of measuring angles in a circular system. Here we are giving the proof for the statement radian is a constant angle. Also, check example questions on expressing radian in the units of the sexagesimal system.

## Prove that Radian is a Constant Angle

A radian is defined as an angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle. Here is the proof for the statement i.e radian is a constant angle.

Let us take a circle with center O and radius r. Also, take any two points on the circle A, B and AB = OA = r, then by the definition ∠AOB = 1 radian.

Produce AO to intersect the circle at point C so that the length of arc ABC is equal to half of the circumference and ∠AOC, the angle at the center subtended by the arc = arc straight angle = two right angles.

$$\frac { arc AB }{ arc ABC }$$ = $$\frac { r }{ ½ x 2πr }$$ = $$\frac { 1 }{ π }$$

$$\frac { ∠AOB }{ ∠AOC }$$ = $$\frac { 1 radian }{ 2 right angles }$$

The arc of a circle is proportional to the angle it subtended at the center of the circle.

Therefore, $$\frac { ∠AOB }{ ∠AOC }$$ = $$\frac { arc AB }{ arc ABC }$$

$$\frac { 1 radian }{ 2 right angles }$$ = $$\frac { 1 }{ π }$$

So, 1 radian = $$\frac { 2 }{ π }$$ right angles

Both 2, π are constants.

Hence, proved.

Also, Check

Question 1:
Express one radian in the units of the sexagesimal system.

= (180 x 7)°/22
= 57° 16′ 22″

Question 2:
Express 3 radians in the units of the sexagesimal system.

= (540 x 7)°/22
= 171°49’5″