Trigonometry Pattern

**xsaiman** · February 28th 2011, 05:06 AM

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

**Soroban** · February 28th 2011, 06:29 AM

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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