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3.5
Derivatives of Trigonometric
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Quick Review
1. Convert 135 degrees to radians.
2. Convert 1.7 radians to degrees.
 
3. Find the exact value of sin   without a calculator.
3
4. State the domain and the range of the cosine function.
5. State the domain and the range of the tangent function.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 3- 2
Quick Review
6.
7.
If sin a   1, what is cos a ?
If tan a   1, what are two possible values of sin a ?
1  cos h
sin 2 h

.
h
h 1  cos h 
8.
Verify the identity:
9.
Find an equation of the line tangent to the curve
y  2 x 3  7 x 2  10 at the point  3,1 .
10. A particle moves along a line with velocity
v  2t 3  7t 2  10 for time t  0. Find the acceleration
of the particle at t  3.
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Slide 3- 3
Quick Review Solutions
1. Convert 135 degrees to radians.
2. Convert 1.7 radians to degrees.
3
 2.356
4
97.403
3
 
3. Find the exact value of sin   without a calculator.
2
3
4. State the domain and the range of the cosine function.
Domain: all reals
Range: [-1,1]
5. State the domain and the range of the tangent function.
k
Domain: x 
Range: all reals
 k odd integer 
2
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Slide 3- 4
Quick Review Solutions
6.
If sin a   1, what is cos a ?
7.
If tan a   1, what are two possible values of sin a ?
0

1
2
sin 2 h
1  cos h
.

h 1  cos h 
h
8.
Verify the identity:
9.
1  cos h
and use the identity 1  cos 2 h  sin 2 h
1  cos h
Find an equation of the line tangent to the curve
Multiply by
y  2 x 3  7 x 2  10 at the point  3,1 . y 12 x  35
10. A particle moves along a line with velocity
v  2t 3  7t 2  10 for time t  0. Find the acceleration
12
of the particle at t  3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 3- 5
What you’ll learn about

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Derivative of the Sine Function
Derivative of the Cosine Function
Simple Harmonic Motion
Jerk
Derivatives of Other Basic Trigonometric Functions
… and why
The derivatives of sines and cosines play a key role in
describing periodic change.
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Slide 3- 6
Graph y = sin(x) and its derivative on the
same coordinate grid. Any conjecture as to
dy/dx?
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Slide 3- 7
Graph y = cos(x) and its derivative on the
same coordinate grid. Any conjecture as to
dy/dx?
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Slide 3- 8
Derivative of the Sine Function
The derivative of the sine is the cosine.
d
sin x  cos x
dx
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Slide 3- 9
Derivative of the Cosine Function
The derivative of the cosine is the negative of the sine.
d
cos x   sin x
dx
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Slide 3- 10
dy
Determine
.
dx
y  3sin x
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Slide 3- 11
dy
Determine
.
dx
1
y   5sin x
x
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Slide 3- 12
dy
Determine
.
dx
x2
y  cos x 
sin x
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Slide 3- 13
dy
Determine
.
dx
y  tan x
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Slide 3- 14
dy
Determine
.
dx
y  sec x
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Slide 3- 15
Derivative of the Other Basic
Trigonometric Functions
d
tan x  sec 2 x
dx
d
cot x   csc 2 x
dx
d
sec x  sec x tan x
dx
d
csc x   csc x cot x
dx
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 3- 16
Example Finding the Derivative of the
Sine and Cosine Functions
Find the derivative of
sin x
.
 cos x  2 
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dy
.
dx
y  2sin x  tan x
Determine
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Determine
dy
.
dx
y  x sec x
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Simple Harmonic Motion
The motion of a weight bobbing up and down on the end of a
string is an example of simple harmonic motion.
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Slide 3- 20
Example Simple Harmonic Motion
A weight hanging from a spring bobs up and down with position


function s  3sint s in meters, t in seconds . What are its velocity
and acceleration at time t ?
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A body is moving in simple harmonic motion with position
function s = f(t). Determine the body’s velocity and
acceleration at time t.
s = 1- 4 cos t
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Slide 3- 22
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
A body is moving in simple harmonic motion with position
function s = f(t). Determine the body’s velocity and
acceleration at time t = π/4.
s = 1- 4 cos t
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 3- 24
Jerk
Jerk is the derivative of acceleration. If a body's position at
time t is
da d 3 s
j t    3 .
dt dt
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A body is moving in simple harmonic motion with position
function s = f(t). Determine the body’s jerk at time t.
s = 1 + 2 cos t
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Slide 3- 26
A body is moving in simple harmonic motion with position
function s = f(t). Determine the body’s velocity and
acceleration at time t = π/4.
s = sin t + cos t.
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Example Derivative of the Other Basic
Trigonometric Functions
Find the equation of a line tangent to y  x cos x at x 1.
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Slide 3- 28
Example Derivative of the Other Basic
Trigonometric Functions
y  x cos x
y  .3012 x  .841
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Slide 3- 29
Find equations for the lines that are tangent and
normal to the graph of y = sec x at x = π/4.
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Slide 3- 30
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Show the graphs of y = tan x and y = cot x have
no horizontal tangents.
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Find the points on the curve y = tan x,
-π/2 < x < π/2, where the tangent is parallel
to the line y = 2x.
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Find y” if y = θ tan θ.
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d 999
Determine 999 (cos x)
dx
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Use the identity cos 2x = cos2 x – sin2 x to find
the derivative of cos 2x. Express the derivative
in terms of sin 2x.
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Which of the following gives y” for
y = cos x + tan x?
2
A. -cos x + 2sec x tan x
B. cos x + 2sec2 x tan x
C. -sin x+ sec2 x
D. -cos x + sec2xtanx
2
E. cos x + sec xtanx
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Assignment
pages 146 – 147,
# 1 – 37 odds and 43
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