DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
We worked hard to show that the derivative of the sine function is the cosine function. (See the page "Derivative of the Sine Function.") Having done this hard work, we can now differentiate the cosine function using these two trigonometric identities.
- cos(x) = sin(x +
/2)
- cos(x + /2) = -sin(x)
Where do THOSE come from?
We differentiate the cosine function as follows. Notice the application of the chain rule in the second step.
| Dx(cos(x)) | = | Dx(sin(x + /2) |
| = | cos(x + /2) Dx(x + /2) |
| = | -sin(x) (1) = -sin(x) |
There are really just six basic trigonometric functions, all contructed from the sine and cosine functions. Aside from sine and cosine, the other four basic trigonometric functions are the secant, cosecant, tangent, and cotangent functions.
Example. Differentiate the tangent function.
Solution.
| Dx(tan(x)) |
= |
|
| = |
| [Dx(sin(x))]cos(x) - sin(x)[Dx(cos(x))] |
|
| cos2(x) |
|
| = |
| cos2(x) + sin2(x) |
|
| cos2(x) |
|
| = |
|
| = | sec2(x) |
Example. Differentiate the secant function.
Solution.
| Dx(sec(x)) |
= |
|
| = |
| [Dx(1)]cos(x) - (1)[Dx(cos(x))] |
|
| cos2(x) |
|
| = |
|
| = |
|
| = | sec(x)tan(x) |
It is an exercise in the use of the quotient rule to differentiate the cosecant and cotangent functions. We leave this to you. Here is a summary of the derivatives of the six basic trigonometric functions.
Derivatives of the Trigonometric Functions
| Dx(sin(x)) = cos(x) |
Dx(cos(x)) = -sin(x) |
| Dx(sec(x)) = sec(x)tan(x) |
Dx(csc(x)) = -csc(x)cot(x) |
| Dx(tan(x)) = sec2(x) |
Dx(cot(x)) = -csc2(x) |
/Stage6/Lesson/trigDeriv.html
© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson