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Geometry Level 3

$\sum_{k=1}^3 \cos^2 \left( (2k-1) \dfrac \pi{12} \right)$

If the summation above can be expressed as $\dfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

1 solution

Alex G
May 28, 2016

$\cos^2{(\dfrac{\pi}{12})}+ \cos^2{(\dfrac{\pi}{4})} + \cos^2{(\dfrac{5\pi}{12})}$

$\cos^2 x= \dfrac{1+\cos{2x}}{2}$

$\dfrac{1+\cos \dfrac{\pi}{6} }{2}+ \dfrac{1}{2}+ \dfrac{1+\cos \dfrac{5\pi}{6} }{2}$

$\dfrac{1+\dfrac{\sqrt{3}}{2}}{2} + \dfrac{1}{2} + \dfrac{1-\dfrac{\sqrt{3}}{2}}{2}$

$\dfrac{3}{2}$

$\boxed{5}$

very nice solution. Alex G.

Ramiel To-ong - 4 years ago