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4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 183
3.4 Derivatives of Trigonometric Functions 183
3.4 Derivatives of Trigonometric Functions
Many of the phenomena we want information about are approximately periodic (electro-
magnetic fields, heart rhythms, tides, weather). The derivatives of sines and cosines play a
key role in describing periodic changes. This section shows how to differentiate the six ba-
sic trigonometric functions.
Derivative of the Sine Function
To calculate the derivative of ƒsxd = sin x, for x measured in radians, we combine the lim-
its in Example 5a and Theorem 7 in Section 2.4 with the angle sum identity for the sine:
sin sx + hd = sin x cos h + cos x sin h.
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 184
184 Chapter 3: Differentiation
If thenƒsxd = sin x,
ƒ ¿sxd = lim ƒsx + hd - ƒsxd
h:0 h
= lim sin sx + hd - sin x Derivative definition
h:0 h
= lim ssin x cos h + cos x sin hd - sin x Sine angle sum identity
h:0 h
= lim sin x scos h - 1d + cos x sin h
h:0 h
= lim asin x # cos h - 1b + lim acos x # sin hb
h:0 h h:0 h
= sin x # lim cos h - 1 + cos x # lim sin h
# h:0 h# h:0 h
= sin x 0 + cos x 1 Example 5(a) and
= cos x. Theorem 7, Section 2.4
The derivative of the sine function is the cosine function:
d ssin xd = cos x.
dx
EXAMPLE 1 Derivatives Involving the Sine
(a) y = x2 - sin x:
dy d
= 2x - Asin xB Difference Rule
dx dx
= 2x - cos x.
(b) y = x2 sin x:
dy 2 d
= x Asin xB + 2x sin x Product Rule
dx dx
= x2 cos x + 2x sin x.
(c) y = sin x:
x
x # d Asin xB - sin x # 1
dy dx
= 2 Quotient Rule
dx x
= x cos x - sin x.
x2
Derivative of the Cosine Function
With the help of the angle sum formula for the cosine,
cos sx + hd = cos x cos h - sin x sin h,
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 185
3.4 Derivatives of Trigonometric Functions 185
we have
d scos xd = lim cossx + hd - cos x Derivative definition
y dx h:0 h
1 y cos x
= lim scos x cos h - sin x sin hd - cos x Cosine angle sum
x h:0 h identity
–
0
–1 cos xscos h - 1d - sin x sin h
y' = lim h
' h:0
y –sin x
1 # cos h - 1 # sin h
x = lim cos x h - lim sin x h
– h:0 h:0
0
–1 # cos h - 1 # sin h
= cos x lim h - sin x lim h
FIGURE 3.23 The curve y¿=-sin x as # h:0 # h:0
the graph of the slopes of the tangents to = cos x 0 - sin x 1 Example 5(a) and
the curve y = cos x. =-sin x. Theorem 7, Section 2.4
The derivative of the cosine function is the negative of the sine function:
d scos xd =-sin x
dx
Figure 3.23 shows a way to visualize this result.
EXAMPLE 2 Derivatives Involving the Cosine
(a) y = 5x + cos x:
dy d d
= s5xd + Acos xB Sum Rule
dx dx dx
= 5 - sin x.
(b) y = sin x cos x:
dy d d
= sin x Acos xB + cos x Asin xB Product Rule
dx dx dx
= sin xs-sin xd + cos xscos xd
= cos2 x - sin2 x.
(c) y = cos x :
1 - sin x
A1 - sin xB d Acos xB - cos x d A1 - sin xB
dy dx dx
= 2 Quotient Rule
dx s1 - sin xd
= s1 - sin xds-sin xd - cos xs0 - cos xd
2
s1 - sin xd
1 - sin x 2 2
= 2 sin x + cos x = 1
s1 - sin xd
= 1 .
1 - sin x
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 186
186 Chapter 3: Differentiation
Simple Harmonic Motion
The motion of a body bobbing freely up and down on the end of a spring or bungee cord is
an example of simple harmonic motion. The next example describes a case in which there
–5 are no opposing forces such as friction or buoyancy to slow the motion down.
EXAMPLE 3 Motion on a Spring
0 Rest
position A body hanging from a spring (Figure 3.24) is stretched 5 units beyond its rest position
and released at time to bob up and down. Its position at any later time t is
t = 0
5 Position at s = 5 cos t.
t 0
s What are its velocity and acceleration at time t?
FIGURE 3.24 A body hanging from Solution We have
avertical spring and then displaced Position: s = 5 cos t
oscillates above and below its rest position. Velocity: y = ds = d s5 cos td =-5 sin t
Its motion is described by trigonometric dt dt
functions (Example 3). dy d
Acceleration: a = dt = dt s-5 sin td =-5 cos t.
s, y Notice how much we can learn from these equations:
1. As time passes, the weight moves down and up between s =-5 and s = 5 on the
y –5 sin t s 5 cos t s-axis. The amplitude of the motion is 5. The period of the motion is 2p.
2. Thevelocityy =-5 sin tattainsitsgreatestmagnitude,5,whencos t = 0,asthe graphs
show in Figure 3.25. Hence, the speed of the weight, ƒ y ƒ = 5 ƒ sin t ƒ , is greatest when
t cos t = 0,thatis,whens = 0(the rest position). The speed of the weight is zero when
0 3 2 5 sin t = 0.Thisoccurswhens = 5 cos t =;5,attheendpointsoftheintervalofmotion.
2 2 2
3. The acceleration value is always the exact opposite of the position value. When the
weight is above the rest position, gravity is pulling it back down; when the weight is
below the rest position, the spring is pulling it back up.
4. The acceleration, is zero only at the rest position, where and
FIGURE 3.25 The graphs of the position a =-5 cos t, cos t = 0
the force of gravity and the force from the spring offset each other. When the weight is
and velocity of the body in Example 3. anywhere else, the two forces are unequal and acceleration is nonzero. The accelera-
tion is greatest in magnitude at the points farthest from the rest position, where
cos t =;1.
EXAMPLE 4 Jerk
The jerk of the simple harmonic motion in Example 3 is
j = da = d s-5 cos td = 5 sin t.
dt dt
It has its greatest magnitude when not at the extremes of the displacement but
sin t =;1,
at the rest position, where the acceleration changes direction and sign.
Derivatives of the Other Basic Trigonometric Functions
Because sin x and cos x are differentiable functions of x, the related functions
tan x = sin x , cot x = cos x , sec x = 1 , and csc x = 1
cos x sin x cos x sin x
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
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