Homework Help: Simplifying multiple trig functions into a single trig function for physics II

1. Jan 19, 2012

Blueban

1. The problem statement, all variables and given/known data

It has been a while since I have really been involved in trig seriously, But I felt it appropriate to go in this particular forum because in my classes from years back "precal" was the title associated with trig (:

The Problem:

sin(X) / sin(X/2) ----->Somehow simplifies into-----> 2*cos(x/2)

I realize this is probably a basic skill I should have, but until this problem I have never really had issues with any of the math in my physics classes, But perhaps I am just looking in the wrong places for the "How" of this situation.

2. Relevant equations

I am 100% sure of the accuracy of this relation, since it was derived from wolfram and then used to answer a refraction physics question which came back correct. I would just like to know where I can look so I can learn how this simplification is possible that I may do this on my own without help.

Thanks!

3. The attempt at a solution

I tried looking at relevant trig Identities...but none of them seem to apply here..seems there is some *tricky* stuff here as my calculus teachers always said haha d:

2. Jan 19, 2012

Curious3141

http://www.sosmath.com/trig/douangl/douangl.html

Covers the double-angle formulae. From these, the half-angle formulae can be simply derived (can you see how?). Scroll down till you see the half-angle formula for sine.

3. Jan 19, 2012

samona

you would need to do some algebraic manipulation. I'll start it out for you...

2cos(x/2) = 2 * ((1 + cosx)/2)^1/2 (half angle identity)
= 4(1+cosx)/2 by squaring.

Then multiply top and bottom by (1-cosx) and reduce to get

(2(1-(cos x)^2) / (1-cosx)

You will eventually get (sinx)^2/(1/2-1/2cosx) and then you will take the root of top and bottom and use the half angle identity to arrive at your answer.

4. Jan 19, 2012

Dick

You can do this a lot of ways, but probably the easiest is to write sin(x)=sin(2*(x/2)). Then use sin(2*a)=2*sin(a)*cos(a).