CK-12-Calculus

http://www.ck12.org Chapter 2. Derivatives

2.4 Derivatives of Trigonometric Functions

Learning Objectives

A student will be able to:

• Compute the derivatives of various trigonometric functions.

If the anglehis measured in radians,
limh→ 0 sinhh=1 and limh→ 01 −cosh h= 0.
We can use these limits to find an expression for the derivative of the six trigonometric functions sinx,cosx,tanx,secx,cscx,
and cotx. We first consider the problem of differentiating sinx, using the definition of the derivative.

d

dx[sinx] =hlim→ 0

sin(x+h)−sinx

h

Since

sin(α+β) =sinαcosβ+cosαsinβ.

The derivative becomes

d

dx[sinx] =limh→ 0

sinxcosh+cosxsinh−sinx

h

=limh→ 0

[

sinx

(cosh− 1

h

)

+cosx

(sinh

h

)]

=−sinx·hlim→ 0

( 1 −cosh

h

)

+cosx·hlim→ 0

(sinh

h

)

=−sinx·( 0 )+cosx·( 1 )

=cosx.

Therefore,

d

dx[sinx] =cosx.

It will be left as an exercise to prove that

d

dx[cosx] =−sinx.