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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

218 Chapter 2 • The Derivative and Its Properties

2.5 The Derivative of the Trigonometric
Functions

OBJECTIVE When you finish this section, you should be able to:
1 Differentiate trigonometric functions (p. 218)
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1 Differentiate Trigonometric Functions

To find the derivatives of y = sin x and y = cos x, we use the limits

sin θ cos θ − 1
lim = 1 and lim = 0
θ→0 θ θ→0 θ

that were established in Section 1.4.

THEOREM Derivative of y = sin x
The derivative of y = sin x is y = cos x. That is,
′
d
′
y = sin x = cos x
dx

Proof
sin(x + h) − sin x
y = lim The definition of a derivative (Form 2)
′
h→0 h
sin x cos h + sin h cos x − sin x
NEED TO REVIEW? The trigonometric
= lim sin(A + B) = sin A cos B + sin B cos A
functions are discussed in Section P.6, h→0 h
pp. 55–61. Trigonometric identities are
discussed in Appendix A.4, pp. A-35 sin x cos h − sin x sin h cos x
= lim Rearrange terms.
to A-38. +
h→0 h h
cos h − 1 sin h

= lim sin x · + · cos x Factor.
h→0 h h
cos h − 1 sin h

= lim sin x lim + lim cos x lim Use properties of limits.
h→0 h→0 h h→0 h→0 h
cos θ − 1 sin θ
= sin x · 0 + cos x · 1 = cos x lim = 0; lim = 1
θ→0 θ θ→0 θ

d
The geometry of the derivative sin x = cos x is shown in Figure 30. On the graph
dx
of f (x) = sin x, horizontal tangent lines are marked as well as the tangent lines that have
slopes of 1 and −1. The derivative function is plotted on the second graph, and those
points are connected with a smooth curve.

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