Thread: Nature of Unit Circle

There is a proof of the addition formula for cosines cos(a-b) that uses a unit circle and the law of cosines to arrive at the law. My question is does this proof only work on the unit circle? In the proof, you substitute 1^2 for two sides of the triangle because they are radii and the radius is 1 unit long in this case. Later in the proof you use sin^2 + cos^2 =1 and everything works out fine. But what if the radius was 2? Wouldn't the proof fail?

Originally Posted by zg12

There is a proof of the addition formula for cosines cos(a-b) that uses a unit circle and the law of cosines to arrive at the law. My question is does this proof only work on the unit circle? In the proof, you substitute 1^2 for two sides of the triangle because they are radii and the radius is 1 unit long in this case. Later in the proof you use sin^2 + cos^2 =1 and everything works out fine. But what if the radius was 2? Wouldn't the proof fail?

In order to have your question answered, I think you are going have to post the proof you are asking about.

One thing I thought of was that you can reproduce the angles in question no matter what the size of the circle, so it's valid to work on the unit circle since all circles are similar?

That works.