This document shows how to use the derivative rules for sin(x), cos(x) and combinations of functions to derive the derivatives for the other trig functions. Some basic trig identities will also be used. The following table summarizes the derivatives of trig functions.
| Function | Its Derivative (with respect to x) | ||
|---|---|---|---|
| 1. | sin(x) | cos(x) | |
| 2. | cos(x) | -sin(x) | |
| 3. | tan(x) | sec2(x) | how to derive |
| 3. | csc(x) | -csc(x)cot(x) | how to derive |
| 3. | sec(x) | sec(x)tan(x) | how to derive |
| 3. | cot(x) | -csc2(x) | how to derive |
| Recall that tan(x) = sin(x)/cos(x). |
| Applying the quotient rule to this gives |
| [cos(x)cos(x) - -sin(x)sin(x)]/cos2, which can be written as |
| [cos2(x) + sin2(x)]/cos2(x). |
| cos2(x) + sin2(x) = 1. This is an important trig identity. |
| Therefore, we have 1/cos2(x), which equals sec2(x) |
| Recall that csc(x) = 1/sin(x). |
| We could apply the quotient rule to this. However, we will use the chain rule version of the power rule to find the derivative of [sin(x)]-1. This gives |
| -sin-2(x)cos(x), which can be rewritten as |
| -[1/sin(x)][cos(x)/sin(x)]. |
| Using basic trig identities, this is equal to -csc(x)cot(x). |
| Recall that sec(x) = 1/cos(x). |
| Thinking of this as [cos(x)]-1 and applying the chain rule version of the power rule gives |
| -cos-2(x)[-sin(x)], which can be rewritten as |
| [1/cos(x)][sin(x)/cos(x)]. |
| Using basic trig identities, this is equal to sec(x)tan(x). |
| Recall that cot(x) = cos(x)/sin(x). |
| Applying the quotient rule to this gives |
| [-sin(x)sin(x) - cos(x)cos(x)]/sin2, which can be written as |
| -[sin2(x) + cos2(x)]/sin2(x). |
| Using the identity cos2(x) + sin2(x) = 1 gives |
| -1/sin2(x), which equals -csc2(x). |