Today is our topic of discussion Circular measurement of angle .
Circular measurement of angle
Definition 1.
By circular system i.e., The measure of an angle in the radian unit is called its radian measure or circular measure.
Suppose ∠MON is a given angle. With centre O we draw a circle of suitable radius OA = r. Suppose the circle intersects the sides OM and ON of the angle at A and B, respectively. So constructed ∠AOB is a centred angle produced by arc AB. Taking an arc AP equal to radius r (arc and radius should have same unit).
Suppose arc AB = s
From proposition 3,
∠MON/∠ AOP = ArcAB/ArcAP = ArcAB/RadiusOA) = s/r
∠MON = s/r x ∠AOP
=s/r x 1 radian = s/r radian
Circular measure of <MON is, where the angle cut are arc s centring its vertex and taking r as radius of the circle.
Proposition 5.
Any arc of length s produces an angle in the centre of the circle of radius r then s = rθ
Particular enunciation:
Let O is the centre and OB = r unit is the radius of the circle ABC, arc AB = s unit and centred ∠AOB = θproduced by the arc AB. It is to prove that, s = rθ
Construction:
Draw the arc BP equal to radius OB at the point B so that it intersects the circle ABC at P. Join O, P.
Proof:
By construction ∠POB = 1^c
We know, centred angle produced by any arc is proportional to arc.
(Arc AB)/(ArcPB) = ∠AOB /∠POB
or, s unit / r unit=θ^c/1^c
or, s/r=θ
s = rθ (Proved)
Relation between the degree and radian (circular) measure
We know from proposition 4,
2 1 radian right angle == П
1 ^ c = 2/pi i.e., right angle. [1 radian = 1 ^ c
therefore 1 right angle = (pi/2) ^ c
or, 90 deg = (pi/2) ^ c
Proposition 6.
Observation :
(i) 90° = 1 right angle= r/2 radian = r/2^c
i.e., 180°= 2 right angle = radian =π^c
(ii) If D °and R^c be the measurement of an angle in sexagesimal and circular system then
D° = (D * pi/180) ^ c = R ^ c
i .e.,D* pi 180 =R
or, D/180 = R/pi
From the above discussion the widely use of the relation between degree and radian measures are given below:
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