Question

period of sin(x+3x+5x)​

Answer 1

SOLUTION

COMPLETE QUESTION

The period of function

$\displaystyle \sf{f(x) = \frac{ \sin x +\sin 3x + \sin 5x +\sin 7x }{ \cos x + \cos 3x +\cos 5x + \cos 7x } }$

$\displaystyle \sf{1) \: \: \frac{\pi}{6} }$

$\displaystyle \sf{2) \: \: \frac{\pi}{3} }$

$\displaystyle \sf{3) \: \: \frac{\pi}{4}}$

$\displaystyle \sf{4) \: \: \frac{\pi}{2} }$

EVALUATION

Here the given function is

$\displaystyle \sf{f(x) = \frac{ \sin x +\sin 3x + \sin 5x +\sin 7x }{ \cos x + \cos 3x +\cos 5x + \cos 7x } }$

We first simplify the given function as below

$\displaystyle \sf{f(x) = \frac{ \sin x +\sin 3x + \sin 5x +\sin 7x }{ \cos x + \cos 3x +\cos 5x + \cos 7x } }$

$\displaystyle \sf{ \implies \: f(x) = \frac{ \sin x +\sin 7x + \sin 3x +\sin 5x }{ \cos x + \cos 7x +\cos 3x + \cos 5x } }$

$\displaystyle \sf{ \implies \: f(x) = \frac{ 2\sin 4x \cos 3x + 2\sin 4x \cos x }{ 2\cos 4x \cos 3x +2\cos 4x \cos x } }$

$\displaystyle \sf{ \implies \: f(x) = \frac{ 2\sin 4x (\cos 3x + \cos x ) }{ 2\cos 4x (\cos 3x + \cos x) } }$

$\displaystyle \sf{ \implies \: f(x) = \frac{ \sin 4x }{ \cos 4x } }$

$\displaystyle \sf{ \implies \: f(x) = \tan 4x}$

We tan 4x is a periodic function

Hence the required period

$\displaystyle \sf{ = \: \: \frac{\pi}{4}}$

FINAL ANSWER

Hence the correct option is

$\displaystyle \sf{3) \: \: \frac{\pi}{4}}$

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