Transcript Slide 1

Homework, Page 356
Convert from DMS to decimal form.
1.
2312
12
2312  23
 28.2
60
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Slide 4- 1
Homework, Page 356
Convert from decimal form to DMS.
5.
21.2
21.2      2112
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Slide 4- 2
Homework, Page 356
Convert from decimal or DMS to radians.
9.
60
   
60 

   3
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Slide 4- 3
Homework, Page 356
Convert from decimal or DMS to radians.
13. 71.72
  
71.72 
  1.2518 rad
  
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Slide 4- 4
Homework, Page 356
Convert from radians to degrees.

17.
6
 180
*
 30
6 
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Slide 4- 5
Homework, Page 356
Convert from radians to degrees.
7

21.
9
7 180
*
 140
9

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Slide 4- 6
Homework, Page 356
Use the appropriate arc length formula to find the missing
information.
r

25. s
? 2 in 25 rad
s
   s  r  2 * 25  50 in
r
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Slide 4- 7
Homework, Page 356
Use the appropriate arc length formula to find the missing
information.
29.
s
r

3m 1m ?
s 3
      3 rad
r 1
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Slide 4- 8
Homework, Page 356
A central angle θ intercepts arcs s1 and s2 on two concentric
circles with radii r1 and r2, respectively. Find the missing
information.
r1
s1
r2
s2
33. 
? 11 cm 9 cm 44 cm ?
s
9
      0.818
r 11
1
1
s s
9 s
  
 s  36 cm
r r
11 44
1
2
2
2
1
2
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Slide 4- 9
Homework, Page 356
37. It takes ten identical pieces to form a circular track
for a pair of toy racing cars. If the inside arc of each
piece is 3.4 inches shorter than the outside arc, what is
the width of the track?
s s 2
s  3.4 s
  

  r  s  3.4   rs
r r 10
r
r
1
2
2
2
2
1
2
1
2
1
2
2
r s  3.4r  rs  3.4r  r s  rs  3.4r   r  r  s
2
2
2
1
2
2
2
2
1
2
2
2
1
2
3.4r
10
r r 
 3.4*
 1.7  r  r  5.341 in
s
2
2
2
1
2
1
2
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Slide 4- 10
Homework, Page 356
41. Which compass bearing is closest to a bearing of
121º?
E  090
SE  135
ESE  112.5 East-southeast is closest to 120
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Slide 4- 11
Homework, Page 356
45. Cathy Nyugen races on a bicycle with 13-inch
radius wheels. When she is traveling at a speed of 44
ft/sec, how many revolutions per minute are her wheels
making?
44 ft 60sec 12 in 1 rev
1
rpm 
 387.848 rpm
sec 1min 1 ft 2 rad 13 in
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Slide 4- 12
Homework, Page 356
49. The captain of the tourist boat Julia follows a 038º
course for 2 miles and then changes course to 047º
for the next 4 miles. Draw a sketch of this trip.
Endpoint
Starting Point
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Slide 4- 13
Homework, Page 356
53. A simple pulley with the given radius r used
to lift heavy objects is positioned 10 feet above
ground level. Given the pulley rotates θº,
determine the height to which the object is
lifted.
a. r = 4 in, θ = 720º
s

   s  r  4*720 *
 16  50.265 in
r
180
b. r = 2 ft, θ = 180º
s

   s  r  2*180 *
 2  6.283 ft
r
180
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Slide 4- 14
Homework, Page 356
57. If horse A is twice as far as horse B from the
center of a merry-go-round, then horse A travels
twice as fast as horse B. Justify your answer.
s
True,   r  s  r  r  2r  s  r   2r
1
1
2
1
2
2
1
since all points on a given radius have the same
angular displacement, horse A will travel twice
as far as horse B for the same angular
displacement, thereby traveling twice as fast.
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Slide 4- 15
Homework, Page 356
61. A bicycle with 26-inch diameter wheels is
traveling at 10 mph. To the nearest whole
number, how many revolutions does each wheel
make per minute?
a. 54
b. 129 c. 259 d. 406 e. 646
s

r
10mi / hr 1rev 5280 ft 12in 1hr

*
*
*
*
 129.283rpm
13in
2 rad
1mi
1 ft 60 min
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Slide 4- 16
Homework, Page 356
Find the difference in longitude between the
given cities.
65. Minneapolis and Chicago
93 16
87 39
5 37
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Slide 4- 17
Homework, Page 356
Assume the cities have the same longitude and
find the distance between them in nautical miles.
69. New Orleans and Minneapolis
44 59
2 

29 57  152  15 
  15.03

60 

15 2
60nm
d
*15.03  902 nautical miles
1
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Slide 4- 18
Homework, Page 356
73. Control tower A is 60 miles east of control
tower B. At a certain time, an airplane bears
340º from control tower A and 037º from control
tower B. Use a drawing to model the exact
location of the airplane.
y
airplane
x
B
A
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Slide 4- 19
4.2
Trigonometric Functions of Acute
Angles
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Quick Review
1. Solve for x.
x
2
3
2. Solve for x.
6
x
3
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Slide 4- 21
Quick Review
3. Convert 9.3 inches to feet.
a
4. Solve for a. 0.45 
20
36
5. Solve for b. 1.72 
b
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Slide 4- 22
Quick Review Solutions
1. Solve for x.
x
2
x  13
3
2. Solve for x.
6
x
x3 3
3
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Slide 4- 23
Quick Review Solutions
3. Convert 9.3 inches to feet. 0.775 feet
a
4. Solve for a. 0.45 
9
20
36
5. Solve for b. 1.72 
900 / 43  20.93
b
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 24
What you’ll learn about




Right Triangle Trigonometry
Two Famous Triangles
Evaluating Trigonometric Functions with a
Calculator
Applications of Right Triangle Trigonometry
… and why
The many applications of right triangle trigonometry
gave the subject its name.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 25
Leading Questions




The functions y = secant x and y = cosecant x
are reciprocal functions.
Given the values of two primary trig functions,
we can calculate the values of the others.
Our left hand provides a key to the basic trig
functions that is always with us.
Given one angle and one side of a right
triangle, we can find the other angle and sides.
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Slide 4- 26
Standard Position
An acute angle θ in standard position, with one
ray along the positive x-axis and the other
extending into the first quadrant.
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Slide 4- 27
Trigonometric Functions
Let  be an acute angle in the right ABC. Then
opp
hyp
sine    sin  
cosecant    csc  
hyp
opp
adj
hyp
cosine    cos  
secant    sec  
hyp
adj
opp
adj
tangent    tan  
cotangent    cot  
adj
opp
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 28
Example Evaluating Trigonometric
Functions of 45º
Find the values of all six trigonometric functions for an
angle of 45º.
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Slide 4- 29
Example Evaluating Trigonometric
Functions of 60º
Find the values of all six trigonometric functions for an
angle of 60º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 30
Example Evaluating Trigonometric for
General Triangles
Find the values of all six trigonometric functions for the angle x
in the triangle shown.
5
7
a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
x
Slide 4- 31
Trigonometric Functions of Five Common
Angles

sin
cos
0
0
2
1
2
2
2
3
2
4
2
4
2
3
2
2
2
1
2
0
2
30
45
60
90
tan
x

0
0
0
3
3
1
3
D.N .E.

30
6

45
4

60
3

90
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
sin
0
cos
1
tan
0
x
0
1
2
2
2
3
2
3
2
2
2
1
2
3
3

1
0
6

1
4

3
3
D.N .E.

2
Slide 4- 32
Trig Value Memory Aid
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Slide 4- 33
Common Calculator Errors When
Evaluating Trig Functions




Using the calculator in the wrong angle mode
(degree/radians)
Using the inverse trig keys to evaluate cot, sec,
and csc
Using function shorthand that the calculator
does not recognize
Not closing parentheses
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 34
Example Evaluating Trigonometric for
General Triangles
Find the exact value of the sine of 60º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 35
Example Solving a Right Triangle
A right triangle with a hypotenuse of 5 inches includes
a 43 angle. Find the measures of the other two angles
and the lengths of the other two sides.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 36
Example Solving a Word Problem
Karen places her surveyor's telescope on the top of a
tripod five feet above the ground. She measures an 8
elevation above the horizontal to the top of a tree that
is 120 feet away. How tall is the tree?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 37
Following Questions




The circular functions get their name from the
fact that we go around in circles trying to
understand them.
Angles are commonly measured counterclockwise from the initial side to the terminal side.
Periods of functions are concerned with the
frequency of their repetition.
A unit circle has a diameter of one and is
located wherever convenient.
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Slide 4- 38
Homework


Review Section 4.2
Page 366, Exercises: 1 – 73 (EOO)
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Slide 4- 39
4.3
Trigonometry Extended: The Circular
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Give the value of the angle  in degrees.
1.  
2
3
2.   

4
Use special triangles to evaluate.
 
3. cot   
 4
 7 
4. cos  

 6 
5. Use a right triangle to find the other five trigonometric
4
functions of the acute angle  given cos  
5
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Slide 4- 41
Quick Review Solutions
Give the value of the angle  in degrees.
2
1.  
120
3
2.   

 45
4
Use special triangles to evaluate.
 
3. cot     1
 4
 7 
4. cos  
  3/2
 6 
5. Use a right triangle to find the other five trigonometric
4
functions of the acute angle  given cos  
5
sec   5 / 4, sin   3 / 5, csc   5 / 3, tan   3 / 4, cot   4 / 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 42
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.
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Slide 4- 43
Initial Side, Terminal Side
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Slide 4- 44
Positive Angle, Negative Angle
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Slide 4- 45
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.
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Slide 4- 46
Example Finding Coterminal Angles
Find a positive angle and a negative angle that are coterminal
with 45.
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Slide 4- 47
Example Finding Coterminal Angles
Find a positive angle and a negative angle that are coterminal
with

6
.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 48
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  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 49
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
( y  0)
( x  0)
( y  0)
Slide 4- 50
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- 51
Example Evaluating More Trig Functions
Find sin 210 without a calculator.
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Slide 4- 52
Example Using one Trig Ratio to Find the
Others
Find sin  and cos  , given tan   4 / 3 and cos   0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 53
Unit Circle
The unit circle is a circle of radius 1 centered at
the origin.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 54
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- 55
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- 56
The 16-Point Unit Circle
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Slide 4- 57