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Math Help - cos sin

  1. #1
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    cos sin

    what is the exact value of 100(cos1+cos2+...cos44)/(sin1+sin2+...sin44)
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  2. #2
    Moo
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    Hello,

    Recall the identities :

    \cos(x)+\cos(y)=2 \cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)

    \sin(x)+\sin(y)=2 \sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)


    Rearrange \cos 1+\cos 2+\dots+\cos 43+\cos 44=(\cos 1+\cos 44)+(\cos 2+\cos 43)+\dots+(\cos 22+\cos 23)
    Using the above formula, this gives :

    =2 \bigg( \cos \frac{45}{2} \cos \frac{43}{2}+\cos \frac{45}{2} \cos \frac{41}{2} + \dots + \cos \frac{45}{2} \cos \frac 12 \bigg)
    =2 \cos \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)


    Similarly, we get :
    \sin 1+\sin 2+\dots+\sin 43+\sin 44=(\sin 1+\sin 44)+(\sin 2+\sin 43)+\dots+(\sin 22+\sin 23)

    =2 \bigg(\sin \frac{45}{2} \cos \frac{43}{2}+\sin \frac{45}{2} \cos \frac{41}{2} + \dots + \sin \frac{45}{2} \cos \frac 12 \bigg)
    =2 \sin \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)



    The expression is now :

    \frac{100(\cos 1+\cos 2+\dots+\cos 44)}{\sin 1+\sin 2+\dots+\sin 44}=100 \cdot \frac{2 \cos \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)}{2 \sin \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)}

    =100 \cdot \frac{\cos \frac{45}{2}}{\sin \frac{45}{2}}=\boxed{100 \cot \frac{45}{2}}
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  3. #3
    Senior Member vincisonfire's Avatar
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    Brilliant.
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