Derivatives of Trigonometric Functions

DERIVATIVE OF THE SINE FUNCTION

We start with the sine function and calculate a change in “y” caused by a change in “x” as

And recalling the equation for the sine function for the sum of two angles, we can write

And we can multiply the following expression by the same term on the numerator and denominator

And remembering the Pythagorean Theorem, we can simplify further

Then inserting this term into that for the change in “y” we have

Then the derivative operation for the sine function is

And from simple geometric considerations, the limit of the sine function divided by the change in “x” is

So that finally we have

DERIVATIVE OF THE COSINE FUNCTION

In a similar manner, we can calculate the change in “y” for a change in “x” for the cosine function as

And the equation for the cosine function for the sum of two angles we have

And we can substitute the expression above to get

The derivative operation for the cosine function is

which, as above, is finally given as