Rational values in a trigonometric function?

Geometry Level 5

Find the sum of all rational numbers q q between 0 and 2 (inclusive) such that cos ⁡ q π \cos{q\pi} is also rational.


The answer is 9.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Guilherme Niedu
Apr 25, 2016

c o s ( q ) cos(q) is only rational for q = k ⋅ π 3 q = \frac{k\cdot\pi}{3} or q = k ⋅ π 2 q = \frac{k\cdot\pi}{2} . This leaves the following possibilites for q q in the given interval:

{ 0 , 1 3 , 1 2 , 2 3 , 1 , 4 3 , 3 2 , 5 3 , 2 } \{0, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, 1, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, 2\}

Which sum up to 9 \fbox{9}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You did a typo in your solution, the answer is 9 9 . Also, can you prove your first statement?

Sharky Kesa - 5 years, 1 month ago

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There is a Niven's theorem for Sin so it is also true for Cos.

Niranjan Khanderia - 5 years, 1 month ago

Corrected while you were commenting! Hehe

About the proof, cosine can only be rational if equal to 0 , ± 1 0, \pm 1 or ± 1 2 \pm \frac{1}{2}

Guilherme Niedu - 5 years, 1 month ago

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Not necessarily. E.g.

cos ⁡ ( arccos ⁡ 1 3 ) = 1 3 \cos({\arccos{\dfrac{1}{3}}})=\dfrac{1}{3}

which has a RHS that is clearly rational.

Sharky Kesa - 5 years, 1 month ago

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