Page:Calculus Made Easy.pdf/190

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

Then we shall have

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

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

Similarly, it follows that

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,