First slide

Extreme values and Periodicity of Trigonometric functions

Question

$What is the fundamental period of f (x) = \frac{\sin x + \sin 3 x}{\cos x + \cos 3 x} ?$

Moderate

A

$π$
B

$\frac{π}{2}$
C

$2 π$
D

$3 π$

Solution

$Given that f (x) = \frac{\sin x + \sin 3 x}{\cos x + \cos 3 x}$
$Clearly f (π + x) = \frac{\sin (π + x) + \sin (3 (π + x))}{\cos (π + x) + \cos (3 (π + x))}$
$= \frac{- \sin x - \sin 3 x}{- \cos x - \cos 3 x}$
$= \frac{\sin x + \sin 3 x}{\cos x + \cos 3 x}$
$= f (x)$
$And f (2 π + x) = \frac{\sin (2 π + x) + \sin (3 (2 π + x))}{\cos (2 π + x) + \cos (3 (2 π + x))}$
$= \frac{\sin x + \sin 3 x}{\cos x + \cos 3 x}$
$= f (x)$
$∴ π, 2 π are periods of f (x)$
$∴ π is the fundamental period of f (x)$

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