Simplifying multiple trig functions into a single trig function for physics II

[Blueban]

Homework Statement

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.

Homework 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!

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:

 

[Curious3141]

Homework Statement

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.

Homework 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!

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:

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.

 

[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.

 

[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).