Math Is Fun Forum

Core2Student

Guest

Radians and Proof Question

Hey, I'm stuck on this question, I'm not even sure where to start, please could you give me some advice on how to go around this?

A chord AB divides a circle with centre O into 2 regions whose areas are in the ratio 2:1. If angle OAB=θradians, show that:

3θ-3sinθ=2π

Thank you in advance.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Radians and Proof Question

Draw a diagram. Draw a radius from O to A and from O to B. Let the radius be of length r.

The area of the larger part is equal to the area of the larger sector + the area of the triangle AOB.

The area of the smaller part is equal to the area of the smaller sector  - the area of the triangle AOB.

Remember that the area of a sector is

and the area of a triangle is

(where a and b = r here).

Also note that the angle of the large sector = 2pi - theta.

Core2Student

Guest

Re: Radians and Proof Question

I get the larger region to equal:

[r²sinθ+r²(2π-θ)]/2

And the smaller region to equal:

[r²θ-r²sinθ]/2

Which means that:

[r²sinθ+2πr²-r²θ]/2 = r²θ-r²sinθ

=> r²sinθ+2πr²-r²θ = 2r²θ-2r²sinθ
=> 3r²sinθ+2πr²-3r²θ = 0
=> ÷r²] 3sinθ+2π-3θ = 0
=> 3θ-3sinθ = 2π

Yay! Thank you! ^^

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Radians and Proof Question

No problem