The Student Room Group

C2 Trig

A circle, centre O, has two radii OA and OB. The line AB divides the circle into two regions whose areas are in the ratio 3:1. If the angle AOB is theta (radians), show that:
theta - sintheta = pi/2

Well, the area is evidently 1/4 (pi*r^2).
In class we were told area = r^2(theta-sintheta).
Using this we get 1/4(pi*r^2) = r^2(theta-sintheta) so 1/4pi = (theta-sintheta), i.e. theta-sintheta = pi/4 not pi/2.... what have I done wrong? :confused:
Reply 1
dinkymints
A circle, centre O, has two radii OA and OB. The line AB divides the circle into two regions whose areas are in the ratio 3:1. If the angle AOB is theta (radians), show that:
theta - sintheta = pi/2

Well, the area is evidently 1/4 (pi*r^2).
In class we were told area = r^2(theta-sintheta).
Using this we get 1/4(pi*r^2) = r^2(theta-sintheta) so 1/4pi = (theta-sintheta), i.e. theta-sintheta = pi/4 not pi/2.... what have I done wrong? :confused:



You have r^2(theta)/2 as the area of the sector and r^2(theta)/2 - (area of triangle AOB) = area of segment.

Therefore; r^2(theta)/2 - r^2sin(theta)/2 = 1/4(pi.r^2)

So that; 1/2[theta - sin(theta)] = pi/4

Hence, theta - sin(theta) = pi/2

Ben
Reply 2
Ben.S.
You have r^2(theta)/2 as the area of the sector and r^2(theta)/2 - (area of triangle AOB) = area of segment.

Therefore; r^2(theta)/2 - r^2sin(theta)/2 = 1/4(pi.r^2)

So that; 1/2[theta - sin(theta)] = pi/4

Hence, theta - sin(theta) = pi/2

Ben


Thanks... probably being stupid here but why r^2(theta)/2?
Reply 3
dinkymints
Thanks... probably being stupid here but why r^2(theta)/2?

Because that's the formula for the area of a sector. If you imagine a circle with radius r, its area will be pi.r^2. Yo can think of this as a sector with theta = 2pi. Therefore, the area of the sector = r^2(theta)/2.

Ben
Reply 4
Ben.S.
Because that's the formula for the area of a sector. If you imagine a circle with radius r, its area will be pi.r^2. Yo can think of this as a sector with theta = 2pi. Therefore, the area of the sector = r^2(theta)/2.

Ben


Thanks :smile: