Download Slide 4- 1 Homework, Page 401

Homework, Page 401
Identify the graph of each function.
1. Graphs of one period of csc x and 2csc x are shown.

The graph of csc x is in blue,
y


and the graph of 2csc x is
in red.

x








Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 1
Homework, Page 401
Describe the graph of the function in terms of a basic trigonometric
function. Locate the vertical asymptotes and graph two periods of
the function.
5.
y
y  tan 2 x
y  tan 2 x  p 


2
Vertical asymptotes at
2n  1 

x
,n 
4
any integer

x






Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 2
Homework, Page 401
Describe the graph of the function in terms of a basic
trigonometric function. Locate the vertical asymptotes and graph
two periods of the function.
y

9. y  2cot 2 x
y  2cot 2 x  p 


2
Vertical asymptotes at
2n
x
, n  any integer
4

x









Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 3
Homework, Page 401
Match the trigonometric function with its graph. Then give Xmin
and Xmax for the viewing window in which the graph is shown.
13. y  2tan x
y  2 tan x matches
graph (a) which has an
Xmin of   and an Xmax
of 
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Slide 4- 4
Homework, Page 401
Analyze each function for Domain, range, continuity, increasing
or decreasing behavior, symmetry, boundedness, extrema,
asymptotes, and end behavior.
17.
y  cot x
Domain : x : x  n , where n is any integer
Range : y : y   or  ,   ; continuous on domain;
decreasing on domain; symmetrical about the origin;
unbounded; no extrema; no horizontal asymptotes;
vertical asymptotes at x  n ; lim cot x and
x 
lim cot x do not exist
x 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 5
Homework, Page 401
Describe the transformations required to obtain the graph of the
given function from a basic trigonometric function.
21. y  3tan x
To obtain the graph of y  3 tan x from the graph
of y  tan x, apply a vertical stretch of 3.
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Slide 4- 6
Homework, Page 401
Describe the transformations required to obtain the graph of the
given function from a basic trigonometric function.
25. y  3cot 1 x
2
1

y  3cot x  p   2
1
2
2
1
To obtain the graph of y  3cot x from the graph
2
of y  cot x, apply a horizontal stretch of 2, a vertical
stretch of 3, and reflect about the x -axis.
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Slide 4- 7
Homework, Page 401
Solve for x in the given interval, using reference triangles in the
proper quadrants.
29. sec x  2,
0 x 
2
1
1

 2  cos x   x 
cos x
2
3
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Slide 4- 8
Homework, Page 401
Solve for x in the given interval, using reference triangles in the
proper quadrants.
33. csc x  1,
2  x  5
2
1
 1  sin x  1  x    5
2
2
sin x
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Slide 4- 9
Homework, Page 401
Use a calculator to solve for x in the given interval.
37. cot x  0.6,
3
2
 x  2
1
tan x 
0.6
x  1.030  2  5.253
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Slide 4- 10
Homework, Page 401
41.
The figure shows a unit circle and an angle t whose
terminal side is in Quadrant III.
y
x^2 + y^2 = 1
P1 (-a, -b)
t
t-pi
x
P2 (a, b)
(a) If the coordinates of P2 are (a, b), explain why the
coordinates of point P1 on the circle and the terminal side of the
angle t – π are (-a, -b).
The line connecting P1 and P2 is a straight-line passing through
the origin, so the points at which the line intersects the unit circle
are reflections about the origin.
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Slide 4- 11
Homework, Page 401
b
41.
(b) Explain why tan t  .
a
b
tan t  because the definition of the tangent of an
a
angle is opposite over adjacent and the opposite side
of the triangle has measure b and the adjacent side
has measure a.
(c) Find tan t – π and show that tan (t) = tan (t – π).
b b
tan  t    
  tan  t 
a a
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Slide 4- 12
Homework, Page 401
41.
.
(d) Explain why the period of the tangent function is π.
The tangent function has a period of  because the
as shown in the example above, its values repeat
after  radians.
(e) Explain why the period of the cotangent function is π.
If the tangent function has a period of  , its reciprocal
function cotangent must also have a period of  radians.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 13
Homework, Page 401
45.
The Bolivar Lighthouse is located on a small island 350 ft
from the shore of the mainland.
350 ft
x
d
(a) Express the distance d as a function of the angle x.
350
350
cos x 
d 
 d  350sec x
d
cos x
(b) If x = 1.55 rad, what is d?
d  350sec x  350sec1.55  16,831.108 ft
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Slide 4- 14
Homework, Page 401
Find approximate solutions for the equation in the interval   x  
49. sec x  5cos x
x  2.034, 1.107,1.107,2.034
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Slide 4- 15
Homework, Page 401
53. The graph of y = cot x can be obtained by a
horizontal shift of
(a) – tan x
(b) – cot x
(c) sec x
(d) tan x
(e) csc x
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Slide 4- 16
Homework, Page 401
Graph both f and g in the [–π, π] by [–10, 10] viewing window.
Estimate values in the interval [–π, π] for which f > g.
57. f  x   5sin x and g  x   cot x
f  g on  0.439,0 
 0.439,  
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Slide 4- 17
Homework, Page 401
61.
Write csc x as a horizontal translation of sec x.

3 


csc x  sec  x   or csc x  sec  x 

2
2




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Slide 4- 18
Homework, Page 401
65. A film of liquid in a thin tube has surface tension γ
given by   0.5h gr sec where h is the height of liquid
in the tube, ρ is the density of the liquid, g = 9.8 m/sec2 is
the acceleration due to gravity, r is the radius of the tube,
and φ is the angle of contact between the tube and the
liquid’s surface. Whole blood has a surface tension of
0.058 N/m and a density of 1050 kg/m3. Suppose the
blood rises to a height of 1.5 m in a capillary blood vessel
of radius 4.7 x 10–6 m. What is the contact angle between
the capillary vessel and the blood surface?
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Slide 4- 19
Homework, Page 401
65. Cont’d

2
sec  

0.5h  gr h  gr
sec  
sec  

2 0.058kg / sec 2




3
2
6
1.5
m
1050
kg
/
m
9.8
m
/
sec
4.7

10
m


2  0.058 
1.51050  9.8  4.7 10
6


 1.599
1
cos  
   51.290
1.599
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Slide 4- 20
4.6
Graphs of Composite Trigonometric
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
State the domain and range of the function.
1. f ( x)  -3sin 2 x
2. f ( x) | x | 2
3. f ( x)  2 cos 3 x
4. Describe the behavior of y  e as x  .
-3 x
5. Find f g and g f , given f ( x)  x  3 and g ( x)  x
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 22
Quick Review Solutions
State the domain and range of the function.
1. f ( x)  -3sin 2 x Domain:  ,   Range:  3,3
2. f ( x) | x | 2
Domain:  ,   Range:  2,  
3. f ( x)  2 cos 3 x Domain:  ,   Range:  2, 2
4. Describe the behavior of y  e as x  .
-3 x
lim e
x 
3 x
0
5. Find f g and g f , given f ( x)  x  3 and g ( x )  x
2
f g  x  3; g f  x  3
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 23
What you’ll learn about



Combining Trigonometric and Algebraic Functions
Sums and Differences of Sinusoids
Damped Oscillation
… and why
Function composition extends our ability to model
periodic phenomena like heartbeats and sound waves.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 24
Leading Questions





The absolute value of a periodic function is a
periodic function.
Adding a linear function to a periodic function
yields a new periodic function.
Adding sinusoids of different periods will
yield a new sinusoid.
Adding sinusoids of different periods will
yield a periodic function.
Damped oscillation refers to a condition where
the amplitude of a function varies.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 25
Example Combining the Cosine Function
with x2
Graph y   cos x  and state whether the function
2
appears to be periodic.
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Slide 4- 26
Example Combining the Cosine Function
with x2
 
Graph y  cos x 2 and state whether the function
appears to be periodic.
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Slide 4- 27
Absolute Values of Trigonometric
Functions
The absolute value of a trig function plots as a
periodic function.
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Slide 4- 28
Sums That Are Sinusoidal Functions
If y  a sin(b( x  h )) and y  a cos(b( x  h )), then
1
1
1
2
2
2
y  y  a sin(b( x  h ))  a cos(b( x  h )) is a
1
2
1
1
2
2
sinusoid with period 2 / | b|.
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Slide 4- 29
Sums That Are Not Sinusoidal Functions
If y  a sin(b( x  h )) and y  f ( x) where f ( x) is not
1
1
1
2
a sin(b( x  h )) or a cos(b( x  h )), but another
2
2
2
2
trigonometric function, then y  y is a periodic
1
2
function, but not a sinusoid.
If y  f ( x) is not a trigonometric function, then y  y
2
1
2
is neither periodic nor sinusoidal.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 30
Example Identifying a Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  3cos x  5sin x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 31
Example Identifying a Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  cos3x  sin 5x
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Slide 4- 32
Example Identifying a Non-Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  3x  sin 5x
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Slide 4- 33
Damped Oscillation
The graph of y  f ( x) cos bx (or y  f ( x) sin bx) oscillates
between the graphs of y  f ( x ) and y   f ( x ). When this
reduces the amplitude of the wave, it is called damped
oscillation. The factor f ( x) is called the damping factor.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 34
Example Working with Damped
Oscillation
The oscillations of a spring subject to friction
are modeled by the equation y  0.43e cos1.8t .
0.55 t
 a  Graph y and its two damping curves in the same
viewing window for 0  t  12.
b
Approximately how long does it take for the spring
to be damped so that  0.2  y  0.2?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 35
Following Questions




Inverse trig function is just another name for a
reciprocal function of a trig function.
Arccosine (x) and cos –1 (x) are the same thing.
All inverse trig functions have restricted
domains.
All inverse trig functions have restricted
ranges.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 36
Homework



Homework Assignment #31
Read Section 4.7
Page 411, Exercises: 1 – 93 (EOO)
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Slide 4- 37
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- 39
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- 40
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.
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Slide 4- 41
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- 42
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- 43
Example Evaluate sin-1x Without a
Calculator
 1
Find the exact value without a calculator: sin   
 2
1
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Slide 4- 44
Example Evaluate sin-1x Without a
Calculator
   
Find the exact value without a calculator: sin  sin    .
  10  
1
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Slide 4- 45
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- 46
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,  .
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Slide 4- 47
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- 48
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- 49
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- 50
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- 51
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.
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Slide 4- 52
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.
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Slide 4- 53
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
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Slide 4- 54