Inspired by Sandeep Bhardwaj problem

Geometry Level 4

$\large f(x) = \dfrac{\sin(x) + \sin(3x) + \sin(5x) + \ldots + \sin(40103x)}{\cos(x) + \cos(3x) + \cos(5x) + \ldots + \cos(40103x)}$

Fundamental period of $f(x)$ can be represented as $\dfrac{a\pi}{b}$ , where $a, b$ are coprime numbers. Find $a + b$ .

Inspiration

The answer is 20053.

1 solution

Aditya Sky
Mar 23, 2016

The given series simplifies to $\tan(20052\cdot x)$ . We know that $\tan(x)$ is a periodic function with fumdamental period $\pi$ , so, $\tan(20052\cdot x)$ is a periodic function with fundamental period $\frac{1}{20052} \cdot \pi$ . Hence, $a\,+\,b\,=\,1\,+\,20052\,=\,20053$

Inspired by Sandeep Bhardwaj problem

The answer is 20053.

1 solution

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