Here we will discuss about radian is a constant angle. Let O be the centre of a circle and radius OR = r. If we take an arc AB = OA = r, then by definition, ∠AOB =1 radian.
Let AO be produced to meet the circle at the point C. Then the length of the arc ABC half the circumference and ∠AOC, the angle at the centre subtended by this arc = a straight angle = two right angles.
Now if we take the ratio of the two arcs and that of the two angles, we have
arc AB/arc ABC = r/(1/2 × 2∙π∙r) = 1/ π
∠AOB/∠AOC = 1 radian/2 right angles
But in geometry, we can show that an arc of a circle is proportional to the angle it subtends at the centre of the circle.
Therefore, ∠AOB/∠AOC = arc AB/arc ABC
or, 1 radian/2 right angles = 1/π
Therefore, 1 radian = 2/π right angles
This is constant as both 2 right angles and π are constants.
The approximate value of π is taken as 22/7 for calculation
π radian =
2 right angles
If we express one radian in the units of sexagesimal system, we will get
1 radian =
(180 × 7°)/22
57° 16’ 22” (approx.)