# [SOLVED] half-angle identity problem

• Jan 1st 2006, 09:08 AM
cherica148
[SOLVED] half-angle identity problem
Hey,
I'm doing a problem in which we are supposed to solve for cos7.5 degrees using the half-angle identities.

I got to +/- square root of (1+cos15)/2, and can't figure out how to find cos15(without just putting it into a calculator) so that I can solve the problem. Did I set this up correctly? If so, then how do I find the cosine of this angle? Thanks!
• Jan 1st 2006, 09:53 AM
CaptainBlack
Quote:

Originally Posted by cherica148
Hey,
I'm doing a problem in which we are supposed to solve for cos7.5 degrees using the half-angle identities.

I got to +/- square root of (1+cos15)/2, and can't figure out how to find cos15(without just putting it into a calculator) so that I can solve the problem. Did I set this up correctly? If so, then how do I find the cosine of this angle? Thanks!

Since $\cos (30)=\frac{\sqrt 3}{2}$ you can use the half angle formula again to find $\cos (15)$.

RonL
• Jan 1st 2006, 10:31 AM
ticbol
Quote:

Originally Posted by cherica148
Hey,
I'm doing a problem in which we are supposed to solve for cos7.5 degrees using the half-angle identities.

I got to +/- square root of (1+cos15)/2, and can't figure out how to find cos15(without just putting it into a calculator) so that I can solve the problem. Did I set this up correctly? If so, then how do I find the cosine of this angle? Thanks!

Yes, you set it up correctly.
You used the trig identity,
cos(A/2) = +,-sqrt[(1 +cosA)/2] ----------***

So from 7.5deg you are now at 15deg.
You need to come up to 30, 45, 60, or 90 degrees so that you don't need to use a calculator.

15deg is half of 30deg, so use again the half-angle cosine function to continue your setup.

cos(7.5deg)
= +,-sqrt[{1 +cos(15deg)}/2]
= +,-sqrt[{0.5 +0.5cos(15deg)}]
= +,-sqrt[{0.5 +0.5(+,-sqrt[(1 +cos(30deg))/2])}]
= +,-sqrt[{0.5 +,-0.5sqrt[(1 +cos(30deg))/2]}]
You can use 0.866 for cos(30deg) here, but if you are solving for the exact cos(7.5deg), then
= +,-sqrt[{0.5 +,-0.5sqrt[(1 +sqrt(3)/2)/2]}]

Zeez, this problem sure like square roots. You are treading dangerous water here. Very easy to make mistake in the simplification.

= +,-sqrt[{0.5 +,-0.5sqrt[(2 +sqrt(3))/(2*2)]}]
= +,-sqrt[{0.5 +,-(0.5/2)sqrt[2 +sqrt(3)]}]
= +,-sqrt[{0.5 +,-(0.25)sqrt[2 +sqrt(3)]}]
= +,-sqrt[{2(0.25) +,-(0.25)sqrt[2 +sqrt(3)]}]
= +,-(0.5)sqrt[{2 +,-sqrt[2 +sqrt(3)]}]

That is it. Cannot simplify it any further.

If you are not allowed to use the calculator even for the square roots and arithmetic, then you are in trouble. I guess.