Download Slide 4- 1 Homework, Page 411

Homework, Page 411
Graph the function from -2π ≤ x ≤ 2π. State whether or not the
function appears to be periodic.
1.
f  x    sin x 
2
f (x) is periodic
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 1
Homework, Page 411
Graph the function from -2π ≤ x ≤ 2π. State whether or not the
function appears to be periodic.
5.
f  x   x cos x
f (x) is not periodic
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 2
Homework, Page 411
Verify algebraically that the function is periodic and determine its
period graphically. Sketch the graph showing two periods.
9.
f  x   cos 2 x
f  x  2    cos  x  2  
  cos  x  

2
y


x
2

 f  x






f  x  is periodic
From the graph, the period is  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 3
Homework, Page 411
State the range and domain of the function and sketch a graph
showing four periods.
13. y  cos 2 x
Domain:x : x 
Range : y : 0  y  1

y


x







Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 4
Homework, Page 411
State the range and domain of the function and sketch a graph
showing four periods.
17. y   tan 2 x
2n  1 

Domain:x : x 
, n any integer
2
Range : y : y  0

y


x










Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 5
Homework, Page 411
The graph of the function oscillates between two parallel lines.
Find equations for the lines and graph the lines and the function.
21.
y  2  0.3x  cos x
y  2  0.3 x  cos x
1  cos x  1
y  3  0.3 x
y  1  0.3 x

y




x








Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 6
Homework, Page 411
Determine whether f (x) is a sinusoid.
25. f  x   2cos  x  sin  x
y

f  x  is a sinusoid because
cos  x and sin  x have the
same period.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

x







Slide 4- 7
Homework, Page 411
Find a, b, and h so that f (x) ≈ a sin (b(x – h)
29. f  x   2sin 2 x  3cos 2 x
f  x   a sin b  x  h 

a  3.5

b2

h


6
y
x






f  x   3.5sin 2  x  
6

    




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 8
Homework, Page 411
Find a, b, and h so that f (x) ≈ a sin (b(x – h)
33.
f  x   2cos x  sin x
f  x   a sin b  x  h 


a  2.2






b 1
h

3


f  x   2.2sin  x  
3










Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y
x
    


Slide 4- 9
Homework, Page 411
The function is periodic, but not a sinusoid. Find the period
graphically and sketch one period)
37. f  x   cos3x  4sin 2 x

y


p  2








   

x













Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 10
Homework, Page 411
Match the function with its graph.
41. f  x   3cos 2 x  cos3x
c.
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Slide 4- 11
Homework, Page 411
Tell whether the function exhibits damped oscillation. If so,
identify the damping factor and tell whether the damping occurs
as x → 0 or x → ∞.
45.
f  x   5 cos1.2 x
There is no damping. The damping function has a
constant amplitude.

y



x









Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 12
Homework, Page 411
Graph both f and plus or minus its damping factor in the same
viewing window. Describe the behavior of f for f > 0. What is
the end behavior of f?
49. f  x   1.2 x cos 2 x
maximum f  x  decreases as x increases
lim f  x   0
x 
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Slide 4- 13
Homework, Page 411
Find the period and graph the function over two periods.
53. f  x   sin 3x  2cos 2 x

p  2
y



x









Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 14
Homework, Page 411
Graph f over [-4π, 4π]. Determine whether the function is
periodic and, if it is, state the period.
57.
1
f  x   sin x  2
2







p  2
  







Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y
x
       
Slide 4- 15
Homework, Page 411
Graph f over [-4π, 4π]. Determine whether the function is
periodic and, if it is, state the period.
61. f  x   1 x  cos 2 x
2
f  x  is not periodic.







  







Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y
x
       
Slide 4- 16
Homework, Page 411
Find the domain and range of the function.
65. f  x   x  cos x
Domain:x : x 
Range:y : y  1













  

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y
x
       
Slide 4- 17
Homework, Page 411
Find the domain and range of the function.
69. f  x   sin x
Domain:x : x 
Range:y : 0  y  1
y


x


   




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 18
Homework, Page 411
73. Example 3 shows that the function f  x   sin 3 x
is periodic. Explain whether you think that f  x   sin x3
is periodic.
The function f  x   sin x is not periodic because the
while x changes at a uniform yrate, x3 does not
3


x


   




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 19
Homework, Page 411
Match the function with its graph and state the viewing window.
77.
y  cos x  sin 2 x  cos3x  sin 4 x
d.
 2 , 2  by  4, 4
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Slide 4- 20
Homework, Page 411
81. The function f  x   sin x is periodic. Justify your
answer.
False. The function sin x is an odd function, the stated
function is an even function, that is not periodic.
 4 , 4  by  2, 2
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Slide 4- 21
Homework, Page 411
85. The function f  x   x  sin x is
a. discontinuous
b. bounded
c. even
d. odd
e. periodic
 4 , 4  by  12,12
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 22
Homework, Page 411
Predict what the graph will look like. Graph the function in one
or more viewing windows, determine the main features, draw a
summary sketch. Where applicable, name the period, amplitude,
domain, range, asymptotes, and zeros.
89.
f  x   cos e x
Domain : x : x 
Range : y : 1  y  1
Horizontal asymptote y  1
Zeros: e
x
2n  1 


, n any integer
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
 4 , 4  by  2, 2
Slide 4- 23
Homework, Page 411
Predict what the graph will look like. Graph the function in one
or more viewing windows, determine the main features, draw a
summary sketch. Where applicable, name the period, amplitude,
domain, range, asymptotes, and zeros.
sin x
f  x 
x
Domain : x : x  0
Range : y : 0.217  y  1
Horizontal asymptote y  0
Zeros : x  n , n : nonzero integer
93.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
 4 , 4  by  2, 2
Slide 4- 24
4.7
Inverse Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
State the sign (positive or negative) of the sine, cosine, and tangent
in quadrant
1. I
2. III
Find the exact value.
3. cos

6
4
4. tan
3
11
5. sin 
6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 26
Quick Review Solutions
State the sign (positive or negative) of the sine, cosine, and tangent
in quadrant
1. I +,+,+
2. III ,,+
Find the exact value.
3. cos

6
4
4. tan
3
11
5. sin 
6
3/2
3
1/ 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 27
What you’ll learn about




Inverse Sine Function
Inverse Cosine and Tangent Functions
Composing Trigonometric and Inverse Trigonometric
Functions
Applications of Inverse Trigonometric Functions
… and why
Inverse trig functions can be used to solve trigonometric
equations.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 28
Leading Questions




The range for the arccosine function is [– 1, 1]
The range of the arcsine function is [–π/2,π/2]
sec –1 x = cos –1 x –1
sin (sin –1 x) = cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 29
Inverse Sine Function
f  x   sin x


2
x

2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
f  x   sin x
1
1  x  1
Slide 4- 30
Inverse Sine Function (Arcsine Function)
The unique angle y in the interval   / 2,  / 2 such that
sin y  x is the inverse sine (or arcsine) of x, denoted
sin 1 x or arcsin x. The domain of y  sin 1 x is [  1,1] and
the range is   / 2,  / 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 31
Example Evaluate sin-1x Without a
Calculator
 1
Find the exact value without a calculator: sin   
 2
1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 32
Example Evaluate sin-1x Without a
Calculator
   
Find the exact value without a calculator: sin  sin    .
  10  
1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 33
Inverse Cosine (Arccosine Function)
f  x   cos x
0 x 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
f  x   cos x
1
1  x  1
Slide 4- 34
Inverse Cosine (Arccosine Function)
The unique angle y in the interval 0,   such that
cos y  x is the inverse cosine (or arccosine) of x,
1
denoted cos x or arccos x. The domain of
y  cos 1 x is [  1,1] and the range is  0,  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 35
Inverse Tangent Function (Arctangent
Function)
f  x   tan x


2
x

2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
f  x   tan x
1
  x  
Slide 4- 36
Inverse Tangent Function (Arctangent
Function)
The unique angle y in the interval (   / 2,  / 2) such
that tan y  x is the inverse tangent (or arctangent )
of x, denoted tan 1 x or arctan x. The domain of
y  tan 1 x is (-,) and the range is (   / 2,  / 2).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 37
End Behavior of the Tangent Function
Recognizing that the graphs of inverse functions are
reflected about the line y = x, we see that vertical
asymptotes of y = tan x become the horizontal asymptotes
of y = tan–1 x and the range of y = tan x becomes the
domain of y = tan–1 x .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 38
Composing Trigonometric and Inverse
Trigonometric Functions
The following equations are always true whenever
they are defined:



sin sin 1  x   x

tan tan

cos cos 1  x   x
1
 x   x
The following equations are only true for x values
in the "restricted" domains of sin, cos, and tan:
sin 1  sin  x    x
cos 1  cos  x    x
tan 1  tan  x    x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 39
Example Composing Trig Functions with
Arcsine
Compose each of the six basic trig functions with sin 1 x
and reduce the composite function to an algebraic
expression involving no trig functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 40
Example Applying Inverse Trig
Functions
A person is watching a balloon rise straight up from a place
500 ft from the launch point.
a. Write θ as a function of s, the
s
height of the balloon.

500 ft
b. Is the change in θ greater as s changes from 10 ft to 20 ft
or as s changes from 200 ft to 210 ft? Explain.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 41
Example Applying Inverse Trig
Functions
c. In the graph of this relationship, does the x-axis represent
s height and the y-axis represent θ (in degrees) or viceversa? Explain.
0,1500 by 5,80
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 42
Following Questions




The angle of depression tells us how depressing
an equation is relative to an equation of known
depression.
The angle of elevation is measured from the
horizontal upwards.
Simple harmonic motion is repetitive motion
such as that of a pendulum.
The frequency of an object in simple harmonic
motion refers to the number of times it passes
through a given point per unit time.
Slide 4- 43
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Homework




Homework Assignment #32
Review Section 4.7
Page 421, Exercises: 1 – 69 (EOO)
Quiz next time
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Slide 4- 44