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Math Help - [SOLVED] half-angle identity problem

  1. #1
    cherica148
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    [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!
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  2. #2
    Grand Panjandrum
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    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
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  3. #3
    MHF Contributor
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    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.
    Last edited by ticbol; January 1st 2006 at 11:34 AM.
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