Angle

Report
Copyright © 2005 Pearson Education, Inc.
Chapter 1
Trigonometric Functions
Copyright © 2005 Pearson Education, Inc.
1.1 Angles
Objective:
Understand and apply the basic terminology of angles
Warm up :
Define and draw a picture of each of the following terms
Line
Ray
Acute angle
Complementary Angles
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Line Segment
Right angle
Obtuse angle
Supplementary Angles
Basic Terms

Two distinct points determine a line called
line AB.
A

B
Line segment AB—a portion of the line
between A and B, including points A and B.
A

B
Ray AB—portion of line AB that starts at A and
continues through B, and on past B.
A
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B
Slide 1-4
Basic Terms continued

Angle-formed by rotating
a ray around its endpoint.

The ray in its initial
position is called the
initial side of the angle.

The ray in its location
after the rotation is the
terminal side of the
angle.
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Slide 1-5
Basic Terms continued

Positive angle: The
rotation of the terminal
side of an angle
counterclockwise.
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
Negative angle: The
rotation of the terminal
side is clockwise.
Slide 1-6
Types of Angles

The most common unit for measuring angles is
the degree.
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Slide 1-7
Example: Complementary Angles


Find the measure of each angle.
Since the two angles form a right
angle, they are complementary
angles. Thus,
k  20  k  16  90
k +20
k  16
2k  4  90
2 k  86
k  43
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The two angles have measures of
43 + 20 = 63 and 43  16 = 27
Slide 1-8
Example: Supplementary Angles


Find the measure of each angle.
Since the two angles form a straight
angle, they are supplementary
angles. Thus,
6 x  7  3 x  2  180
9 x  9  180
6x + 7
3x + 2
9 x  171
x  19
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These angle measures are
6(19) + 7 = 121 and 3(19) + 2 = 59
Slide 1-9
Degree, Minutes, Seconds

One minute is 1/60 of a degree.
1
1' 
60

or
60'  1
One second is 1/60 of a minute.
1
1
1" 

60 3600
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or
60"  1'
Slide 1-10
Example: Calculations

Perform the calculation.
27 34' 26 52'

Perform the calculation.
72  15 18'
27 34'

 26 52'
53 86'

Write 72 as 71 60'
71 60
Since 86 = 60 + 26, the
sum is written
53
15 18'
 1 26'
56 42'
54 26'
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Slide 1-11
Example: Conversions


Convert to decimal
degrees.
74 12' 18"
12
18

60 3600
 74  .2  .005
74 12' 18"  74 
 74.205


Convert to degrees,
minutes, and seconds
36.624
34.624  34  .624
 34  .624(60')
 34  37.44'
 34  37 ' .44'
 34  37 ' .44(60")
 34  37 ' 26.4"
 34 37 ' 26.4"
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Slide 1-12
Standard Position

An angle is in standard position if its vertex is
at the origin and its initial side is along the
positive x-axis.

Angles in standard position having their terminal
sides along the x-axis or y-axis, such as angles
with measures 90, 180, 270, and so on, are
called quadrantal angles.
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Slide 1-13
Coterminal Angles

A complete rotation of a ray results in an angle
measuring 360. By continuing the rotation,
angles of measure larger than 360 can be
produced. Such angles are called coterminal
angles.
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Slide 1-14
Example: Coterminal Angles




Find the angles of smallest possible positive
measure coterminal with each angle.
a) 1115
b) 187
Add or subtract 360 as may times as needed to
obtain an angle with measure greater than 0 but
less than 360.
o
o
o
a) 1115  3(360 )  35
b) 187 + 360 = 173
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Slide 1-15
Homework

Page 7 # 14 - 42
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Slide 1-16
1.2
Objective:
Compare Angle Relationships and to
identify similar triangles and calculate
missing sides and angles.
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Warm up: Use the graph at the right to find the following
1. Name a pair of vertical angles.
2. Line a and b are what kind of lines.
3. Name a pair of alternate interior angles.
1
2
a
4. Name a pair of alternate exterior angles
5. Name a pair of corresponding angles.
3
b
4
5
6
7
6. Find the measure of all the angles.
Copyright © 2005 Pearson Education, Inc.
8
Angles and Relationships
q
m
n
Name
Angles
Rule
Alternate interior angles
4 and 5
3 and 6
Angles measures are equal.
Alternate exterior angles
1 and 8
2 and 7
Angle measures are equal.
Interior angles on the same
side of the transversal
4 and 6
3 and 5
Angle measures add to 180.
Corresponding angles
2 & 6, 1 & 5,
3 & 7, 4 & 8
Angle measures are equal.
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Slide 1-19
Vertical Angles

Vertical Angles have equal measures.
Q
R
M
N

P
The pair of angles NMP and RMQ are vertical
angles.
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Slide 1-20
Parallel Lines


Parallel lines are lines that lie in the same plane
and do not intersect.
When a line q intersects two parallel lines, q, is
called a transversal.
Transversal
q
m
parallel lines
n
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Slide 1-21
Example: Finding Angle Measures

Find the measure of each
marked angle, given that
lines m and n are parallel.
(6x + 4)
(10x  80)
m


n


The marked angles are
alternate exterior angles,
which are equal.
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6 x  4  10 x  80
84  4 x
21  x
One angle has measure
6x + 4 = 6(21) + 4 = 130
and the other has measure
10x  80 = 10(21)  80 =
130
Slide 1-22
Angle Sum of a Triangle

The sum of the measures of the angles of any
triangle is 180.
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Slide 1-23
Example: Applying the Angle Sum

The measures of two of
the angles of a triangle
are 52 and 65. Find the
measure of the third
angle, x.

Solution
52  65  x  180
117  x  180
x  63
65
x

The third angle of the
triangle measures 63.
52
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Slide 1-24
Types of Triangles: Angles
Copyright © 2005 Pearson Education, Inc.
Slide 1-25
Types of Triangles: Sides
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Slide 1-26
Conditions for Similar Triangles

Corresponding angles must have the same
measure.

Corresponding sides must be proportional.
(That is, their ratios must be equal.)
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Slide 1-27
Example: Finding Angle Measures

Triangles ABC and DEF
are similar. Find the
measures of angles D
and E.


D
Since the triangles are
similar, corresponding
angles have the same
measure.
Angle D corresponds to
angle A which = 35
A
112
35
F
C
112
33
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E

Angle E corresponds to
angle B which = 33
B
Slide 1-28
Example: Finding Side Lengths

Triangles ABC and DEF
are similar. Find the
lengths of the unknown
sides in triangle DEF.

32 64

16 x
32 x  1024
x  32
D
A

16
112
35
64
F
32
C
112
33
48
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To find side DE.
B
E
To find side FE.
32 48

16 x
32 x  768
x  24
Slide 1-29
Example: Application

A lighthouse casts a
shadow 64 m long. At the
same time, the shadow
cast by a mailbox 3 feet
high is 4 m long. Find the
height of the lighthouse.

The two triangles are
similar, so corresponding
sides are in proportion.
3 x

4 64
4 x  192
x  48
3

4
x
The lighthouse is 48 m
high.
64
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Slide 1-30
Homework

Page 14-16 # 3-13
odd, 25-35 odd, 45-56
odd
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Slide 1-31
1.3
Objective: To understand and
apply the 6 trigonometric
functions
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Warm up

In the figure below, two similar triangles are
present. Find the value of each variable.
x-2y
5
74
x-5
x+y
10
74
15
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Slide 1-33
Trigonometric Functions

Let (x, y) be a point other the origin on the
terminal side of an angle  in standard position.
The distance from the point to the origin is
r  x 2  y 2 . The six trigonometric functions of 
are defined as follows.
y
sin  
r
r
csc  ( y  0)
y
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x
cos 
r
y
tan   (x  0)
x
r
sec  ( x  0)
x
x
cot  
(y  0)
y
Slide 1-34
Example: Finding Function Values

The terminal side of angle  in standard position
passes through the point (12, 16). Find the
values of the six trigonometric functions of
angle .
r  x 2  y 2  122  162
(12, 16)
16
 144  256  400  20

12
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Slide 1-35
Example: Finding Function Values
continued

x = 12
y 16 4
sin   

r 20 5
x 12 3
cos  

r 20 5
y 16 4
tan   

x 12 3
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y = 16
r = 20
r 20 5
csc  

y 16 4
r 20 5
sec  

x 12 3
x 12 3
cot   

y 16 4
Slide 1-36
Example: Finding Function Values

Find the six trigonometric
function values of the
angle  in standard
position, if the terminal
side of  is defined by
x + 2y = 0, x  0.

We can use any point on
the terminal side of  to
find the trigonometric
function values.
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Slide 1-37
Example: Finding Function Values
continued

Choose x = 2
x  2y  0
2  2y  0
2 y  2

y  1
The point (2, 1) lies on
the terminal side, and the
corresponding value of r
is r  22  (1) 2  5.
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
Use the definitions:
y 1 1 5
5
sin   



r
5
5
5 5
x
2
2
5 2 5




r
5
5
5 5
y
1
r
tan    
csc    5
x
2
y
cos 
sec 
r
5

x
2
cot  
x
 2
y
Slide 1-38
Example: Function Values Quadrantal
Angles


Find the values of the six trigonometric functions for an angle
of 270.
First, we select any point on the terminal side of a 270 angle.
We choose (0, 1). Here x = 0, y = 1 and r = 1.
1
sin 270 
 1
1
1
tan 270 
undefined
0
1
sec 270  undefined
0
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0
cos 270   0
1
1
csc 270 
 1
1
0
cot 270   0
1
Slide 1-39
Undefined Function Values

If the terminal side of a quadrantal angle lies
along the y-axis, then the tangent and secant
functions are undefined.

If it lies along the x-axis, then the cotangent and
cosecant functions are undefined.
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Slide 1-40
Commonly Used Function Values

sin 
cos 
tan 
cot 
sec 
csc 
0
0
1
0
undefined
1
undefined
90
1
0
undefined
0
undefined
1
180
0
1
0
undefined
1
undefined
270
1
0
undefined
0
undefined
1
360
0
1
0
undefined
1
undefined
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Slide 1-41
Homework

Page 25 # 18-46 even
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Slide 1-42
1.4
Objective: to apply the
definitions of the trigonometric
functions
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Warm-up







What is the reciprocal of
2/3?
1 2/5?
0?
Cos 0?
Sin 0?
Tan 0?
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Slide 1-44
Reciprocal Identities

1
sin  
csc
1
cos 
sec
1
tan  
cot 
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
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Slide 1-45
Example: Find each function value.


2
cos  if sec  =
3
Since cos  is the
reciprocal of sec 
1
1 3
cos 
 2
sec 3 2

15
sin  if csc   
3
1
3
sin  

15
15

3
3 • 15
3 • 15


15
15 • 15

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15
5
Slide 1-46
Signs of Function Values
 in
sin  cos 
tan 
cot 
sec 
csc 
Quadrant
I
+
+
+
+
+
+
II
+




+
III


+
+


IV

+


+

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Slide 1-47
Example: Identify Quadrant




Identify the quadrant (or quadrants) of any angle
 that satisfies tan  > 0, cot  > 0.
tan  > 0 in quadrants I and III
cot  > 0 in quadrants I and III
so, the answer is quadrants I and III
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Slide 1-48
Ranges of Trigonometric Functions




For any angle  for which the indicated functions
exist:
1. 1  sin   1 and 1  cos   1;
2. tan  and cot  can equal any real number;
3. sec   1 or sec   1 and
csc   1 or csc   1.
(Notice that sec  and csc  are never between
1 and 1.)
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Slide 1-49
Identities

Pythagorean
sin 2   cos 2   1,
tan 2   1  sec 2  ,
1  cot 2   csc 2 
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
Quotient
sin 
 tan 
cos 
cos 
 cot 
sin 
Slide 1-50
Example: Other Function Values



Find sin and cos if tan  = 4/3 and  is in
quadrant III.
Since is in quadrant III, sin and cos will both
be negative.
sin and cos must be in the interval [1, 1].
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Slide 1-51
Example: Other Function Values
continued

We use the identity tan 2   1  sec 2 
tan 2   1  sec 2 
2
4
2

1

s
ec

 
3
16
 1  sec 2 
9
25
 sec 2 
9
5
  sec
3
3
  cos 
5
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Since sin 2   1  cos 2  ,
 3
sin 2   1    
 5
9
sin 2   1 
25
16
sin 2  
25
4
sin   
5
2
Slide 1-52
Homework

Page 33-35 # 4-10, 16, 18, 56-62
Copyright © 2005 Pearson Education, Inc.
Slide 1-53

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