If the frequency, or number of periods per second, be denoted by , then , 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 , but to . For this we must resort to the artifice explained in Chapter IX., p. 67 and put
.
Now will obviously be ; so that
.
Similarly, it follows that
.
Second Differential Coefficient of Sine or Cosine.
We have seen that when is differentiated with respect to it becomes ; and that when is differentiated with respect to it becomes ; or, in symbols,