Question

proove that radian is a constant angle !!!

Answer 1

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
=180

If we express one radian in the units of sexagesimal system, we will get

1 radian = 180°/(22/7)

=(180 × 7°)/22

=57° 16’ 22” (approx.)