Posted by Chandler on Monday, December 10, 2007 at 11:36am.
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)]
I need the actual value like 1/2 or sqrt3/2.
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
I was wondering why you were using cosin the problem when the half angle formula for sin is +-sqrt((1-cosx)/2). I can get pi/12, but I can't figure out how to use that answer to get to pi/24
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[1-cos^2(pi/24)]
Ok, so say I use the half angle formula in sin. I get:
sin((pi/6)/2)= sqrt((1-sqrt3/2)/2).
So sin(pi/12)=sqrt(2-sqrt3)/2
I get to here by multiplying the whole equation by (2/2).
So from here, (whew) how to I get to pi/24
I'm confused. I can't seem to make the answer out of what you have.
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[1-cos^(pi/24)]. You have the symbolic expression for cos(pi/24), so you'll get symbolic expression for
sin(pi/24)
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)]