First slide

Functions (XII)

Question

The period of the function f (x) = a sin kx + b cos kx is

Moderate

A

$\frac{2 π}{k}$
B

$\frac{2 π}{|k|}$
C

$\frac{π}{|k|}$
D

None of these

Solution

We have, f (x) = a sin kx + b cos kx
$\begin{array}{l} = \sqrt{a^{2} + b^{2}} (\frac{a}{\sqrt{a^{2} + b^{2}}} \sin kx + \frac{b}{\sqrt{a^{2} + b^{2}}} \cos kx) \\ = \sqrt{a^{2} + b^{2}} (\cos θsin kx + \sin θcos kx) \end{array}$
where $\cos θ = \frac{a}{\sqrt{a^{2} + b^{2}}}$
$= \sqrt{a^{2} + b^{2}} \sin (kx + θ)$
which is a periodic function of period $\frac{2 π}{| k |} .$

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