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 