Let f(x) = sin x + cos(ax) be a periodic func | vedclass

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(C) function $f(x) = \sin(px) + \cos(qx)$ is periodic if and only if the ratio of the periods of the two components is a rational number. The period o

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$a$ is a rational number

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(C) function $f(x) = \sin(px) + \cos(qx)$ is periodic if and only if the ratio of the periods of the two components is a rational number. The period of $\sin x$ is $T_1 = 2\pi$. The period of $\cos(ax)$ is $T_2 = \frac{2\pi}{|a|}$ (for $a 
eq 0$). For the sum to be periodic, the ratio $\frac{T_1}{T_2}$ must be a rational number. $\frac{T_1}{T_2} = \frac{2\pi}{2\pi / |a|} = |a|$. Thus, $|a|$ must be a rational number. Therefore, $a$ must be a rational number.

Let $f(x) = \sin x + \cos(ax)$ be a periodic function. Then:

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