Radian measure [Circular system]

Ask a Question, Get an Answer!

Type your question here

Hundreds of tutors are online and ready to help you right now!

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 180^o

1 radian = 1^c. Thus p^c stands for p radian. In practice, we usually write p only and not p^c. Thus the angle of p always indicates p radian (p^c).

(p is an irrational number)

Some important results

\ 180^o = p radian = 180^o

1 radian = 57.2958^o (approximately)

= 57^o 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 

\ 2p = 360^o

p = 180^o