A radian is the measure of an angle at the centre of a circle subtended by an arc, whose arc length is equal to the radius of the circle.
OR
A radian is the angle formed by a half line that has been rotated about
its end point of a complete
rotation.
Let O be the centre of circle of radius r and if the circular arc AB is equal in length to the radius r of the circle, then by definition AB = 1 radian.
Theorem 1
Statement:
Radian is a constant angle.
Proof:
Let PQR be a circle with centre O and radius r. Consider an arc AB of the circle whose length is equal to radius of the circle. Draw OA and OB and Produce AO to cut the circle at C.
Angle at the centre is proportional to the length of the corresponding
arc thus
or
p radian 180o
1 radian = 1c. Thus pcstands for p radian. In practice, we usually write p only and not pc. Thus the angle of p always indicates p radian (pc).
(p is an irrational number)
Some important results
\ 180o = p radian = 180o
1 radian = 57.2958o(approximately)
= 57o 17' 44.81" (approximately)
= 206265 seconds (approximately)
= 0.01745 radian
Theorem 2
The number of radians in angle subtend by an arc of a circle at the
Let
.
Now draw a circle of radius r which intersect OP at A and OQ at C.
Consider an arc AB on the circle, so that the length of the arc and the length of radius are both equal to r. Draw OB.
Then by definition of radian
Let s = arc length of AC
Since the angle at the centre is proportional to the arc length on which they stand
\ s r q
Hence the magnitude of an angle in circular measure
Relation between degrees and radian
Let a ray revolve about its end
point O and generates a circle of radius r.
Then the radian measure of the angle formed by one complete
revolution of the revolving ray
of a complete
rotation. 
p radian 
180o 
.
Now draw a circle of radius r which intersect OP at A and OQ at C.
r q
revolve about its end
point O and generates a circle of radius r.
p = 180o
