Transcript Slide 1

Homework, Page 366
Find the values of all six trigonometric functions of the angle x.
5
4
1.
x
3
4
3
4
sin x  ;cos x  ; tan x 
5
5
3
5
5
3
csc x  ;sec x  ;cot x 
4
3
4
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Slide 4- 1
Homework, Page 366
Find the values of all six trigonometric functions of the angle x.
7
5.
x
11
c  a  b  7  11  150  5 6
2
2
2
2
7
7 6
11 11 6
7
sin x 

;cos x 

; tan x 
30
30
11
5 6
5 6
5 6
5 6
11
csc x 
;sec x 
;cot x 
7
11
7
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Slide 4- 2
Homework, Page 366
Assume that θ is an acute angle in a right triangle satisfying the
given condition. Evaluate the remaining trigonometric functions.
9. sin   3
7
3
sin    a  c 2  b 2  49  9  2 10
7
2 10
3
3 10
cos 
; tan  

7
20
2 10
7
7
7 10
2 10
csc x  ;sec 

;cot  
3
20
3
2 10
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Slide 4- 3
Homework, Page 366
Assume that θ is an acute angle in a right triangle satisfying the
given condition. Evaluate the remaining trigonometric functions.
13. tan   5
9
5
2
2
tan    c  a  b  25  81  106
9
3 106
9
5 106
5

;cos  

sin  
106
106
106
106
9
106
106
;cot  
;sec 
csc x 
5
9
5
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Slide 4- 4
Homework, Page 366
Assume that θ is an acute angle in a right triangle satisfying the
given condition. Evaluate the remaining trigonometric functions.
17. csc  23
9
23
csc 
 a  c 2  b 2  529  81  448  8 7
9
9
8 7
9
9 7
sin   ;cos 
; tan 

23
23
56
8 7
23 23 7
8 7
sec 

;cot  
56
9
8 7
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Slide 4- 5
Homework, Page 366
Evaluate without using a calculator.
21. cot   
 
6
 
3
cos  
6
 

cot   
 2  3
1
 6  sin   
 
2
6
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Slide 4- 6
Homework, Page 366
Evaluate using a calculator. Given an exact value.
25. sec45
1
2
2 2
sec 45 


 2
2
2
2
2
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Slide 4- 7
Homework, Page 366
Evaluate using a calculator, giving answers to three decimal
places.
29. sin74
sin74  0.961
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Slide 4- 8
Homework, Page 366
Evaluate using a calculator, giving answers to three decimal
places.
 
33. tan  
 12 
 
tan    0.268
 12 
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Slide 4- 9
Homework, Page 366
Evaluate using a calculator, giving answers to three decimal
places.
37. cot 0.89
cot 0.89  0.810
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Slide 4- 10
Homework, Page 366
Without a calculator, find the acute angle θ that satisfies the
equation. Give θ in both degrees and radians.
41. sin   1
2
1

sin      30 
2
6
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Slide 4- 11
Homework, Page 366
Without a calculator, find the acute angle θ that satisfies the
equation. Give θ in both degrees and radians.
45. sec  2
1

sec  2  cos     60 
2
3
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Slide 4- 12
Homework, Page 366
Solve for the variable shown.
49.
15
x
34
15
15
sin 34   x 
 26.824
x
sin 34
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Slide 4- 13
Homework, Page 366
Solve for the variable shown.
53.
35
6
y
6
6
sin 35   y 
 10.461
y
sin 35
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Slide 4- 14
Homework, Page 366
Solve for the variable shown.
y
57.
a

  55a  15.58
c

b
x
a
a
  55a  15.58  cos    c 
 27.163
c
cos 
b
  35  tan    b  a tan   22.251
a
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Slide 4- 15
Homework, Page 366
61. A guy wire from the top of a radio tower forms a
75º angle with the ground at a 55 ft distance from the
foot of the tower. How tall is the tower?
b
  75a  55  tan  
a
b  a tan   55 tan 75  205.263 ft
75
55
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Slide 4- 16
Homework, Page 366
65. A surveyor wanted to measure the length of a lake. Two
assistants, A and C, positioned themselves at opposite ends of the
lake and the surveyor positioned himself 100 feet perpendicular to
the line between the assistants and on the perpendicular line from
the assistant C. If the angle between his lines of sight to the two
assistants is 79º12‘42“, what is the length of the lake?
A
AC AC
  751242  tan  

BC 100
42
12.7
751242  7512 
 75 
 75.212
60
60
AC  100 tan 751242  378.797 ft
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C
75
12'
42"
B
Slide 4- 17
Homework, Page 366
69. Which of the following expressions does not
represent a real number?
a. sin 30º
b. tan 45º
sin 30  0.5; tan 45  1
c. cod 90º
cos90  0;csc90  11
d. csc 90º
csc90  01
e. sec 90º
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Slide 4- 18
Homework, Page 366
73. The table is a simplified trig table. Which column is
the values for the sine, the cosine, and the tangent functions?
Angle
?
?
?
40º
42 º
44 º
0.8391
0.9004
0.9657
0.6428
0.6691
0.6047
0.7660
0.7431
0.7191
46 º
48 º
50 º
1.0355
1.1106
1.1917
0.7191
0.7431
0.7660
0.6047
0.6691
0.6428
The second column is tangent values, because tangent can be greater
than one, the third is sine values, because they are increasing and the
fourth column is cosine values because they are decreasing.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 19
4.3
Trigonometry Extended: The Circular
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about




Trigonometric Functions of Any Angle
Trigonometric Functions of Real Numbers
Periodic Functions
The 16-point unit circle
… and why
Extending trigonometric functions beyond triangle
ratios opens up a new world of applications.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 21
Leading Questions
We may substitute any real number n for θ in any
trig function and find the value of the function.
Cosine is negative in the fourth quadrant.
Coterminal angles have the same measure.
Quadrantal angles have their terminal sides in the
center of the quadrants.
The period of a trig function tells us how often it
takes on identical values.

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Slide 4- 22
Initial Side, Terminal Side
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Slide 4- 23
Positive Angle, Negative Angle
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Slide 4- 24
Coterminal Angles
Two angles in an extended angle-measurement
system can have the same initial side and the
same terminal side, yet have different measures.
Such angles are called coterminal angles.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 25
Example Finding Coterminal Angles
Find a positive angle and a negative angle that are coterminal
with 45.
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Slide 4- 26
Example Finding Coterminal Angles
Find a positive angle and a negative angle that are coterminal
with

6
.
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Slide 4- 27
Example Evaluating Trig Functions
Determined by a Point in Quadrant I
Let  be the acute angle in standard position whose terminal
side contains the point (3,5). Find the six trigonometric functions
of  .
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Slide 4- 28
Trigonometric Functions of any Angle
Let  be any angle in standard position and let P( x, y )
be any point on the terminal side of the angle (except
the origin). Let r denote the distance from P ( x, y ) to
the origin, i.e., let r  x  y . Then
2
y
sin  
r
x
cos 
r
y
tan  
( x  0)
x
2
r
csc  
y
r
sec  
x
x
cot  
y
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( y  0)
( x  0)
( y  0)
Slide 4- 29
Evaluating Trig Functions of a
Nonquadrantal Angle θ
1.
2.
3.
4.
5.
Draw the angle θ in standard position, being careful to place the
terminal side in the correct quadrant.
Without declaring a scale on either axis, label a point P (other than
the origin) on the terminal side of θ.
Draw a perpendicular segment from P to the x-axis, determining
the reference triangle. If this triangle is one of the triangles whose
ratios you know, label the sides accordingly. If it is not, then you
will need to use your calculator.
Use the sides of the triangle to determine the coordinates of point
P, making them positive or negative according to the signs of x
and y in that particular quadrant.
Use the coordinates of point P and the definitions to determine the
six trig functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 30
Signs of Trigonometric Functions
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Slide 4- 31
Reference Angles
The acute angle made by the terminal side of an
angle and the x-axis is called the reference angle.
The absolute value of each trig function is equal to
the absolute value of the same trig function of the
reference angle in the first quadrant. The sign of
the trig function is determined by the quadrant in
which the terminal side lies.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 32
Example Evaluating More Trig Functions
Find sin 210 without a calculator.
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Slide 4- 33
Example Using one Trig Ratio to Find the
Others
Find sin  and cos  , given tan   4 / 3 and cos   0.
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Slide 4- 34
Unit Circle

The unit circle is a circle of radius 1 centered
at the origin.
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Slide 4- 35
Trigonometric Functions of Real Numbers
Let t be any real number, and let P( x, y ) be the point
corresponding to t when the number line is wrapped
onto the unit circle as described above. Then
1
sin t  y
csc t 
( y  0)
y
1
cos t  x
sec t 
( x  0)
x
y
x
tan t 
( x  0)
cot t 
( y  0)
x
y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 36
Periodic Function
A function y  f (t ) is periodic if there is a positive number c such that
f (t  c)  f (t ) for all values of t in the domain of f . The smallest such
number c is called the period of the function.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 37
The 16-Point Unit Circle
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Slide 4- 38
Following Questions





Graphs of the sine function may be stretched
vertically, but not horizontally.
Horizontal stretches of the cosine function are the
result of changes in its period.
Horizontal translations of the sine function are the
result of phase shifts.
Sinusoids are functions whose graphs have the
shape of the sine curve.
Sinusoids may be used to model periodic
behavior.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 39
Homework




Homework Assignment #28
Review Section 4.3
Page 381, Exercises: 1 – 69 (EOO)
Quiz next time
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Slide 4- 40
4.4
Graphs of Sine and Cosine: Sinusoids
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
State the sign (positive or negative) of the function in each quadrant.
1. sin x
2. cot x
Give the radian measure of the angle.
3. 150
4.  135
5. Find a transformation that will transform the graph of y  x to
1
the graph of y  2 x .
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 42
Quick Review Solutions
State the sign (positive or negative) of the function in each quadrant.
1. sin x +,+,, 
2. cot x +,,+, 
Give the radian measure of the angle.
3. 150 5 /6
4.  135
 3 /4
5. Find a transformation that will transform the graph of y  x to
1
the graph of y  2 x . vertically stretch by 2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 43
What you’ll learn about



The Basic Waves Revisited
Sinusoids and Transformations
Modeling Periodic Behavior with Sinusoids
… and why
Sine and cosine gain added significance when
used to model waves and periodic behavior.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 44
Sinusoid
A function is a sinusoid if it can be written in the form
f ( x)  a sin(bx  c)  d where a, b, c, and d are constants
and neither a nor b is 0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 45
Amplitude of a Sinusoid
The amplitude of the sinusoid f ( x)  a sin(bx  c)  d is |a|.
Similarly, the amplitude of f ( x)  a cos(bx  c)  d is |a|.
Graphically, the amplitude is half the distance between the
trough and the crest of the wave.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 46
Example Finding Amplitude
Find the amplitude of each function and use the language of
transformations to describe how the graphs are related.
(a)
y1  sin x
1
(b) y2  2sin x (c) y1  sin x
3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 47
Period of a Sinusoid
The period p of the sinusoid f ( x)  a sin(bx  c)  d is
p  2 / | b | . Similarly, the period of f ( x)  a cos(bx  c)  d
is p  2 / | b | . Graphically, the period is the length of one
full cycle of the wave.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 48
Example Finding Period and Frequency
Find the period and frequency of each function and use the
language of transformations to describe how the graphs are
related.
(a)
y1  sin x (b) y2  2sin  2 x 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
x
(c) y1  3sin
3
Slide 4- 49
Example Horizontal Stretch or Shrink
and Period
 x
Find the period of y  sin   and use the language of
2
transformations to describe how the graph relates to
y  sin x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 50
Frequency of a Sinusoid
The frequency f of the sinusoid f ( x)  a sin(bx  c)  d
is f | b | / 2  1 p . Similarly, the frequency of
f ( x)  a cos(bx  c)  d is f | b | / 2  1 p . Graphically,
the frequency is the number of complete cycles the wave
completes in a unit interval.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 51
Example Combining a Phase Shift with a
Period Change
Construct a sinusoid with period  /3 and amplitude 4
that goes through (2,0).
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Slide 4- 52
Graphs of Sinusoids
The graphs of y  a sin(b( x  h))  k and y  a cos(b( x  h))  k
(where a  0 and b  0) have the following characteristics:
amplitude = |a | ;
2
period =
;
|b|
|b|
frequency =
.
2
When complared to the graphs of y  a sin bx and y  a cos bx,
respectively, they also have the following characteristics:
a phase shift of h;
a vertical translation of k .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 53
Constructing a Sinusoidal Model using
Time
1. Determine the maximum value M and minimum value m.
M m
The amplitude A of the sinusoid will be A 
, and
2
M m
the vertical shift will be C 
.
2
2. Determine the period p, the time interval of a single cycle
of the periodic function. The horizontal shrink (or stretch)
2
will be B 
.
p
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 54
Constructing a Sinusoidal Model using
Time
3. Choose an appropriate sinusoid based on behavior
at some given time T . For example, at time T :
f (t )  A cos( B(t  T ))  C attains a maximum value;
f (t )   A cos( B(t  T ))  C attains a minimum value;
f (t )  A sin( B(t  T ))  C is halfway between a minimum
and a maximum value;
f (t )   A sin( B(t  T ))  C is halfway between a maximum
and a minimum value.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 55
Example Constructing a Sinusoidal
Model
On a certain day, high tide occurs at 7:12 AM and the
water depth is measured at 15 ft. On the same day, low
tide occurs at 1:24 and the water depth measures 8 ft.
(a) Write a sinusoidal function modeling the tide.
(b) What is the approximate depth of water at 11:00 AM?
At 3:00 PM?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 56
Example Constructing a Sinusoidal
Model
(c) At what time did the first low tide occur? The second
high tide?
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Slide 4- 57