2.5, The Chain Rule Applied to Some Trig Functions

We take an extra day on the chain rule to allow students time to assimilate this rule and build up their skills in manipulating it.

When we apply the chain rule to trig and exponential functions, we gain information and understanding.

Once we work out the derivatives of the basic trig functions, sine, tangent and secant, the chain rule gives us a nice way to keep up with the other functions.  Here is how:

If theta is an acute angle of a right triangle, the opposite acute angle is said to be complementary. (In the figure below, the complementary angle of the black angle theta is the blue angle pi/2 - theta.)

The sine of the complementary angle is call the co-sine; the tangent of the complementary angle is call the co-tangent; secant of the complementary angle is call the co-secant -- this explains the syllable "co" in cosine, cotangent and cosecant.

Since the derivative of

pi/2 - theta 

is -1, then by the chain rule, the derivative of

f(pi/2-theta)

is

-f'(pi/2-theta).

Here is a table of the six trig functions, each matched with its complementary function.

And here are the derivatives.

Note how the derivatives listed in the right column are the negatives of the complementary functions of the derivatives given in the left column.