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Math Help - sum of cosine functions

  1. #1
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    sum of cosine functions

    Find the sum of cos 1⁰ + cos 2⁰ + cos 23⁰ +..+ cos 179⁰.

    I know the sum would be zero but I can't show how to get it. Please help me to arrive an algebraic.
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  2. #2
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    \displaystyle \cos(\theta+ \varphi) = 2 \cos\left( \dfrac{\theta + \varphi} {2} \right) \cos\left( \dfrac{\theta - \varphi}{2} \right)

    Take pairs from opposite ends such that \alpha + \varphi = 180^{\circ}


    edit: Removed working because as Archie Meade pointed out in post 3 the difference is not 180. The sum is still 180 though and because each sum is 180 the answer is 0 since we're multiplying by 0.

    Perhaps it would be better to follow the working in post 3 for a more complete solution
    Last edited by e^(i*pi); January 3rd 2011 at 08:00 AM. Reason: See post
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  3. #3
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    Quote Originally Posted by vldo View Post
    Find the sum of cos 1⁰ + cos 2⁰ + cos 23⁰ +…..+ cos 179⁰.

    I know the sum would be zero but I can't show how to get it. Please help me to arrive an algebraic.
    cos(\pi-A)=cos{\pi}cos(-A)-sin{\pi}sin(-A)=-cos(-A)=-cosA

    \Rightarrow\ cos179^o=cos\left(180^o-1^o\right)=-cos1^o

    cos178^o=cos\left(180^o-2^o\right)=-cos2^o

    \Rightarrow\ cos1^o+cos2^o.....+cos179^o=cos1^o+cos179^o+cos2^o  +cos178^o+....+cos89^o+cos91^o+cos90^o

    =cos1^o-cos1^o+cos2^o-cos2^o+......+cos89^o-cos89^o+cos90^o

    =cos90^o


    @ e^(i*pi)

    the difference of the angles is not 180 degrees.
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