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Geometry Level 4

$f(x)=\dfrac{\sin x+\sin 3x+\sin 5x+\sin 7x}{\cos x+\cos 3x+\cos 5x+\cos 7x}$

What is the fundamental period of the function above?

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\dfrac{\pi}{2}

\dfrac{\pi}{4}

2\pi

\pi

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Sandeep Bhardwaj
Dec 6, 2014

$f(x)=\dfrac{sinx+sin3x+sin5x+sin7x}{cosx+cos3x+cos5x+cos7x}$

$\quad = \dfrac{\left(sinx+sin7x \right) + \left(sin3x+sin5x \right)}{\left(cosx+cos7x \right)+ \left( cos3x+cos5x \right)}$

$\quad =\dfrac{2sin4x.cos3x+2.sin4x.cosx}{2cos4x.cos3x+2cos4x.cosx}$

$\quad =\dfrac{2.sin4x \left(cos3x+cosx \right)}{2.cos4x \left( cos3x+cosx \right)}=tan4x$

$\implies f(x)=tan4x$

So the fundamental period is $\dfrac{\pi}{4}$

Actually, $f(x)=\tan 4x$ $\text{ iff } x\notin {P\cup Q}$ , where,

$P=\{(2m+1)\frac{\pi}{4}, m\in \mathbb{Z}\} \text{ and } Q=\{(2n+1)\frac{\pi}{2}, n\in \mathbb{Z}\}$

If $x\in P\cup Q$ , then $f(x)$ becomes undefined.

The answer came out correct because the undefined points are also periodic and at rest of the points, the curve is same as the curve $y=\tan 4x$ whose fundamental period is $\dfrac{\pi}{4}$ .

Prasun Biswas - 6 years, 6 months ago

I just did the same

Aakash Khandelwal - 6 years, 5 months ago

Could you please elaborate on what do you mean by fundamental period? Thanks

Sanchit Ahuja - 6 years, 2 months ago

fundamental period of function $f$ is $p$ if $f(x) = f(x+p)$ , where $p$ is as small as possible. For example: for $f(x) = \sin x$ then $f(x) = f(x+ 2\pi)$ , so $p = 2\pi$ ; for $f(x) = \tan x$ then $f(x) = f(x + \pi)$ , so $p = \pi$ .

Pi Han Goh - 6 years, 2 months ago

How good are your fundamentals?

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