Question about Trig Identities

[Flumpster]

Homework Statement

This isn't really a problem that was assigned to me, (I'm studying independently) I just have a question about the general concept behind some identities.

Homework Equations

sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)

sin(theta) = sin(180-theta)

The Attempt at a Solution

I'm trying to understand 2 things:

Firstly, I've been looking at proofs of the sine addition formula and a lot seem to be based on the proof you see pictured in Wikipedia

(http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Sine)

and it looks to me like this proof would only apply when angles a+b are smaller than 90 degrees. Are these proofs meant to only prove this identity for sums smaller than 90, or do they actually prove it directly for all angles in some way which I'm not seeing?


The second thing is, I know that sin(theta) = sin(180-theta).

If there is a proof of something like the sine addition identity for angle sums smaller than 90, can I use the sin(theta) = sin(180-theta) identity (or any similar identity for values outside the 0-90 degree range) to apply the addition identity proof to larger angles?

(I know that the sine values repeat, I'm not sure if the proofs for one range of values can be applied to another range)

This is probably very basic stuff but this would really help me out a lot! Thanks :)

 

[HallsofIvy]

Yes, that is correct. The given proof applies only when all angles involved lie between 0 and 90 degrees (in radians, 0 to $\pi/2$). If you want to be able to apply sine and cosine to arbitrary numbers, you really need a completely different definition- the common "circle" definition (In fact, in some textbooks, sine and cosine are called "circular functions" rather than "trigonometric functions"). That definition is:
Draw the unit circle on a coordinate system (the circle with center at (0, 0) and radius 1- equivalently, the graph of the relation $x^2+ y^2= 1$). Given real number t, starting at (1, 0), measure around the circumference a distance t (counter-clockwise if t is positive, clock-wise if t is negative). cos(t) and sin(t) are defined to be the x and y coordinates, respectively, of the ending point. One can then use formulas for the distance between points (cos(a), sin(a)), (cos(b), sin(b)) and between (cos(a-b), sin(a-b)) and (1, 0) to get formulas for cos(a- b) and sin(a- b), the change the sign on b to get cos(a+b) and sin(a+b).

 

[Flumpster]

Ok, that makes sense. Thanks a lot for your help! :)