Deriving Sin(a+b) from the Unit Circle: Any Ideas?

[Corky]

How do I derive the formula 'Sin(a+b)=sinacosb + cosasinb' from the unit circle. Any ideas would be appreaciated our study group tried and failed.

 

[Oxymoron]

http://www.clowder.net/hop/cos(a+b).html[/U]

 

[HallsofIvy]

That's hard! (Well, more tedious than hard.)

It's a little easier to prove 'Sin(a-b)=sinacosb - cosasinb'
(and then change the sign on b).

The basic idea is to set up the points whose coordinates are
(cos(a),sin(a)) (i.e. the point a distance a from (0,0) measured along the circle) and (cos(b),sin(b)) and calculate the straight line distance between them (the arc distance, along the circle, is a-b, of course.) Now mark the point whose arc length from (1,0) is also a-b: it's coordinates are (cos(a-b), sin(a-b)) and calculate the straight line distance beween it and (1,0). Since the arclengths are the same, the lengths of these chords are the same. Set the two calculations equal and "grind".

 

[HallsofIvy]

Having actually sat down and done the calculation, I find that my suggestion gives the cos(x+y) and cos(x-y). I'm going to have to think about how to get sin(x+y)!