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Math Help - Trigonometry Pattern

  1. #1
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    Trigonometry Pattern

    Hey guys,
    allright so I have this question for math that my teacher gave me:

    cos 360 = 1
    cos 180 + cos 360= 0
    cos 120 + cos 240 + cos 360 = 0
    cos 90 + cos 180 + cos 270 + cos 360 = 0
    cos 72 + cos 144 + cos 216 + cos 288 + cos 360 = 0

    a) whats the pattern here?
    b) what formula/equation can be made?

    if anyone could help that would be good
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, xsaiman!

    \begin{array}{c}\cos 360 \;=\; 1 \\<br />
\cos 180 + \cos 360\;=\; 0 \\<br />
\cos 120 + \cos 240 + \cos 360 \;=\; 0 \\<br />
\cos 90 + \cos 180 + \cos 270 + \cos 360 \;=\; 0 \\<br />
\cos 72 + \cos 144 + \cos 216 + \cos 288 + \cos 360 \;=\; 0 \\ \vdots \end{array}

    a) What's the pattern here?

    b) What formula/equation can be made?

    We can ignore the first equation.

    We note this pattern . . .

    The second equation has multiples of 180^o.
    The third has multiples of 120^o
    The fourth has multiples of 90^o.
    The fifth has multiple of 72^o.

    \text{The }n^{th}\text{ equation has multiples of }\dfrac{360^o}{n}


    \displaystyle \text{A formula could be: }\;\sum^n_{k=1}\cos\left(\frac{360^o}{n}k\right) \;=\;0\;\text{ for }n \ge 2

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