Radian is a Constant Angle

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

Radian is a Constant Angle

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


Corollary:

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.)

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