Download Trigonometric Functions

1
Trigonometric
Functions
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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1
Trigonometric Functions
1.1 Angles
1.2 Angle Relationships and Similar
Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the
Trigonometric Functions
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1.1 Angles
Basic Terminology ▪ Degree Measure ▪ Standard Position ▪
Coterminal Angles
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1.1
Example 1 Finding the Complement and the Supplement
of an Angle (page 3)
For an angle measuring 55°, find the measure of its
complement and its supplement.
Complement: 90° − 55° = 35°
Supplement: 180° − 55° = 125°
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1.1
Example 2(a) Finding Measures of Complementary and
Supplementary Angles (page 3)
Find the measure of each angle.
The two angles form a right angle, so they are
complements.
The measures of the two angles are
and
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1.1
Example 2(b) Finding Measures of Complementary and
Supplementary Angles (page 3)
Find the measure of each angle.
The two angles form a straight angle, so they are
supplements.
The measures of the two angles are
and
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1.1
Example 3 Calculating with Degrees, Minutes, and
Seconds (page 4)
Perform each calculation.
(a)
(b)
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1.1
Example 4 Converting Between Decimal Degrees and
Degrees, Minutes, and Seconds (page 5)
(a) Convert 105°20′32″ to decimal degrees.
(b) Convert 85.263° to degrees, minutes, and seconds.
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1.1
Example 5 Finding Measures of Coterminal Angles
(page 6)
Find the angles of least possible positive measure
coterminal with each angle.
(a) 1106°
Add or subtract 360° as many times as needed to obtain
an angle with measure greater than 0° but less than 360°.
An angle of 1106° is coterminal with an angle of 26°.
(b) –150°
An angle of –150° is coterminal with an angle of 210°.
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1.1
Example 5 Finding Measures of Coterminal Angles
(cont.)
(c) –603°
An angle of –603° is coterminal with an angle of 117°.
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1.1
Example 6 Analyzing the Revolutions of a CD Player
(page 7)
A wheel makes 270 revolutions per minute. Through
how many degrees will a point on the edge of the
wheel move in 5 sec?
The wheel makes 270 revolutions in one minute or
revolutions per second.
In five seconds, the wheel makes
revolutions.
Each revolution is 360°, so a point on the edge of the
wheel will move
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1.2 Angles
Geometric Properties ▪ Triangles
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1.2
Example 1 Finding Angle Measures (page 11)
Find the measures of angles 1, 2,
3, and 4 in the figure, given that
lines m and n are parallel.
Angles 2 and 3 are interior angles on the same side of the
transversal, so they are supplements.
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1.2
Example 1 Finding Angle Measures (cont.)
Angles 1 and 2 have equal measure because they are
vertical angles, and angles 1 and 4 have equal measure
because they are alternate exterior angles.
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1.2
Example 2 Finding Angle Measures (page 12)
The measures of two of the angles of a triangle are
33° and 26°. Find the measure of the third angle.
The sum of the measures of the angles of a triangle is
360°.
Let x = the measure of the third angle.
The third angle measures 121°.
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1.2
Example 3 Finding Angle Measures in Similar Triangles
(page 13)
In the figure, triangles DEF
and GHI are similar. Find
the measures of angles G
and I.
The triangles are similar, so the corresponding angles
have the same measure.
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1.2
Example 4 Finding Side Lengths in Similar Triangles
(page 14)
Given that triangle MNP
and triangle QSR are
similar, find the lengths of
the unknown sides of
triangle QSR.
The triangles are similar, so the lengths of the
corresponding sides are proportional.
PM corresponds to RQ.
PN corresponds to RS.
MN corresponds to QS.
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1.2
Example 4 Finding Side Lengths in Similar Triangles
(cont.)
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1.2
Example 5 Finding the Height of a Flagpole (page 14)
Joey wants to know the
height of a tree in a park
near his home. The tree
casts a 38-ft shadow at
the same time that Joey,
who is 63 in. tall, casts a
42-in. shadow. Find the
height of the tree.
Let x = the height of the tree
The tree is 57 feet tall.
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1.3 Trigonometric Functions
Trigonometric Functions ▪ Quadrantal Angles
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1.3
Example 1 Finding Function Values of an Angle (page 22)
The terminal side of an angle θ in standard position
passes through the point (12, 5). Find the values of
the six trigonometric functions of angle θ.
x = 12 and y = 5.
13
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1.3
Example 2 Finding Function Values of an Angle (page 22)
The terminal side of an angle θ in standard position
passes through the point (8, –6). Find the values of
the six trigonometric functions of angle θ.
x = 8 and y = –6.
10
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1.3
Example 2 Finding Function Values of an Angle (cont.)
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1.3
Example 3 Finding Function Values of an Angle (page 23)
Find the values of the six trigonometric functions of
angle θ in standard position, if the terminal side of θ is
defined by 3x – 2y = 0, x ≤ 0.
Since x ≤ 0, the graph of the
line 3x – 2y = 0 is shown to the
left of the y-axis.
Find a point on the line:
Let x = –2. Then
A point on the line is (–2, –3).
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1.3
Example 3 Finding Function Values of an Angle (cont.)
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1.3
Example 4(a) Finding Function Values of Quadrantal
Angles (page 25)
Find the values of the six trigonometric functions of a
360° angle.
The terminal side passes
through (2, 0). So x = 2 and
y = 0 and r = 2.
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1.3
Example 4(b) Finding Function Values of Quadrantal
Angles (page 25)
Find the values of the six trigonometric functions of
an angle θ in standard position with terminal side
through (0, –5).
x = 0 and y = –5 and r = 5.
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1.4
Using the Definitions of the
Trigonometric Functions
Reciprocal Identities ▪ Signs and Ranges of Function Values ▪
Pythagorean Identities ▪ Quotient Identities
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1.4
Example 1 Using the Reciprocal Identities (page 29)
Find each function value.
(a) tan θ, given that cot θ = 4.
tan θ is the reciprocal of cot θ.
(b) sec θ, given that
sec θ is the reciprocal of cos θ.
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1.4
Example 2 Finding Function Values of an Angle (page 30)
Determine the signs of the trigonometric functions of
an angle in standard position with the given measure.
(a) 54°
(b) 260°
(c) –60°
(a) A 54º angle in standard position lies in quadrant I, so all its
trigonometric functions are positive.
(b) A 260º angle in standard position lies in quadrant III, so its
sine, cosine, secant, and cosecant are negative, while its
tangent and cotangent are positive.
(c) A –60º angle in standard position lies in quadrant IV, so
cosine and secant are positive, while its sine, cosecant,
tangent, and cotangent are negative.
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1.4
Example 3 Identifying the Quadrant of an Angle (page 31)
Identify the quadrant (or possible quadrants) of an
angle θ that satisfies the given conditions.
(a) tan θ > 0, csc θ < 0
tan θ > 0 in quadrants I and III, while csc θ < 0 in
quadrants III and IV. Both conditions are met only in
quadrant III.
(b) sin θ > 0, csc θ > 0
sin θ > 0 in quadrants I and II, as is csc θ. Both conditions
are met in quadrants I and II.
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1.4
Example 4 Deciding Whether a Value is in the Range of
a Trigonometric Function (page 32)
Decide whether each statement is possible or
impossible.
(a) cot θ = –0.999
(b)cos θ = –1.7 (c)csc θ = 0
(a) cot θ = –0.999 is possible because the range of cot θ
is
(b) cos θ = –1.7 is impossible because the range of cos θ
is [–1, 1].
(c) csc θ = 0 is impossible because the range of csc θ is
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1.4
Example 5 Finding All Function Values Given One Value
and the Quadrant (page 32)
Angle θ lies in quadrant III, and
Find the
values of the other five trigonometric functions.
Since
and y = –8.
and θ lies in quadrant III, then x = –5
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1.4
Example 5 Finding All Function Values Given One Value
and the Quadrant (cont.)
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1.4
Example 6 Finding Other Function Values Given One
Value and the Quadrant (page 34)
Find cos θ and tan θ given that sin θ
cos θ > 0.
and
Reject the negative root.
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1.4
Example 6 Finding Other Function Values Given One
Value and the Quadrant (cont.)
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1.4
Example 7 Using Identities to Find Function Values (page
35)
7
Find sin θ and cos θ given that cot θ  24 and
θ is in quadrant II.
Since θ is in quadrant II, sin θ > 0 and cos θ < 0.
7
24
cot   
 tan  
24
7
tan2   1  sec 2 
2
 24 
2

1

sec

 7 


625
 sec 2 
49
25

 sec 
7
7
cos   
Quad II
25
sin2   1  cos2 
 7 
sin   1   

 25 
2
2
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1.4
Example 7 Finding Other Function Values Given One
Value and the Quadrant (cont.)
 7 
sin   1   

25


2
2
576
sin  
625
24
sin 
25
2
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