Posted by Chandler on Monday, December 10, 2007 at 11:36am.
Find the exact value of sin(pi/24)
So like i need like the value like sqrt3/2 but for pi/24...
also if you could show me the steps so i could figure the rest out, that would be cool

Math  Count Iblis, Monday, December 10, 2007 at 11:45am
Use the formula cos(2x) = 2 cos^2(x)  1
You can rewrite this as:
cos(x) = ±sqrt{[cos(2x) + 1]/2}
Or
cos(x/2) = ±sqrt{[cos(x) + 1]/2}
cos(pi/6) = 1/2 sqrt(3)
Therefore:
cos(pi/12) = sqrt[1/4 sqrt(3) + 1/2]
cos(pi/24) =
sqrt{[sqrt[1/4 sqrt(3) + 1/2] + 1]/2}
>
sin(pi/24) = sqrt[1 cos^2(pi/24)] 
Math  Chandler, Monday, December 10, 2007 at 12:00pm
I need the actual value like 1/2 or sqrt3/2.

Math  Count Iblis, Monday, December 10, 2007 at 12:18pm
The value is in the form of nested square roots as I gave above.
E.g. there also exists an expression for cos(2 pi/17) and sin(2 pi/17), see here:
http://en.wikipedia.org/wiki/Heptadecagon#Heptadecagon_construction
cos(2 pi/n) can be expressed in terms of square roots if n is a power of two times a product of distinct Fermat prime numbers. Fermat prime numbers are prime numbers of the form 2^(2^n) + 1 
Math  Chandler, Monday, December 10, 2007 at 12:27pm
I was wondering why you were using cosin the problem when the half angle formula for sin is +sqrt((1cosx)/2). I can get pi/12, but I can't figure out how to use that answer to get to pi/24

Math  Count Iblis, Monday, December 10, 2007 at 12:30pm
Because the half ange formla of the sin involves the cosine, while the half angle formula for the cosine only involves cosine. The best strategy is thus to compute cos(pi/24) first and then express the sin in terms of the cos using
sin(pi/24) = sqrt[1cos^2(pi/24)] 
Math  Chandler, Monday, December 10, 2007 at 12:35pm
Ok, so say I use the half angle formula in sin. I get:
sin((pi/6)/2)= sqrt((1sqrt3/2)/2).
So sin(pi/12)=sqrt(2sqrt3)/2
I get to here by multiplying the whole equation by (2/2).
So from here, (whew) how to I get to pi/24 
Math  Chandler, Monday, December 10, 2007 at 12:21pm
I'm confused. I can't seem to make the answer out of what you have.

Math  Count Iblis, Monday, December 10, 2007 at 12:28pm
You just compute cos(x/2) (assumed to be positive) from cos(x) using this formula
cos(x/2) = sqrt{[cos(x) + 1]/2}
Take x = pi/6, then cos(pi/6) is known to be 1/2 sqrt(3). The formula gives you cos(pi/12). Then you take x : pi/12, you now now what the cosine is and you work out cos(pi/24). You do all of this symbolically, not numerically, so you get exact expressions.
Then you express sin(pi/24) as
sqrt[1cos^(pi/24)]. You have the symbolic expression for cos(pi/24), so you'll get symbolic expression for
sin(pi/24) 
Math  drwls, Monday, December 10, 2007 at 12:28pm
sin (pi/6)= sin A = = 1/2
You want the sine of 1/4 of that angle.
There is a trig identity that says
sin (1/2)A = sqrt [(1/2)(1  cos A)]
and another that says
cos(1/2)A = sqrt [(1/2)(1 + cos A)]
sin (pi/12) = sqrt [(1/2)(1 cosA)]
= sqrt [0.5(1  (sqrt3)/2)]
You can apply trigonometric identities one more time to get sin (pi/24), but it gets extremely messy. You will need the value of cos pi/12, which is
sqrt [0.5(1 + (sqrt3)/2)]