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170

Calculus Made Easy

If the frequency, or number of periods per second, be denoted by $n$ , then $n={\dfrac {1}{T}}$ , and we may then write:

$\theta =2\pi nt$ .

Then we shall have

$y=\sin 2\pi nt$ .

If, now, we wish to know how the sine varies with respect to time, we must differentiate with respect, not to $\theta$ , but to $t$ . For this we must resort to the artifice explained in Chapter IX., p. 67 and put

${\frac {dy}{dt}}={\frac {dy}{d\theta }}\cdot {\frac {d\theta }{dt}}$ .

Now ${\dfrac {d\theta }{dt}}$ will obviously be $2\pi n$ ; so that

${\begin{aligned}{\frac {dy}{dt}}&=\cos \theta \times 2\pi n\\&=2\pi n\cdot \cos 2\pi nt.\\\end{aligned}}$ .

Similarly, it follows that

${\frac {d(\cos 2\pi nt)}{dt}}=-2\pi n\cdot \sin 2\pi nt$ .

Second Differential Coefficient of Sine or Cosine.

We have seen that when $\sin \theta$ is differentiated with respect to $\theta$ it becomes $\cos \theta$ ; and that when $\cos \theta$ is differentiated with respect to $\theta$ it becomes $-\sin \theta$ ; or, in symbols,

${\frac {d^{2}(\sin \theta )}{d\theta ^{2}}}=-\sin \theta .$