Transcript Angle
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
2k 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
60' 1
or
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.
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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
84 4 x
21 x
n
The marked angles are
alternate exterior angles,
which are equal.
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6 x 4 10 x 80
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
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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
15
5
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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 sec2
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
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Slide 1-53