Transcript 1.1 Angles Basic Terminology Degree Measure

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 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 2 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 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 4)
(a) Convert 105°20′32″ to decimal degrees.
(b) Convert 85.263° to degrees, minutes, and seconds.
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1.2 Angles
Geometric Properties ▪ Triangles
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1.2
Example 1 Finding Angle Measures in Similar Triangles
(page 14)
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 2 Finding Side Lengths in Similar Triangles
(page 15)
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 3 Finding Side Lengths in Similar Triangles
(cont.)
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1.2
Example 5 Finding the Height of a Flagpole (page 14)
Samir 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 as Samir,
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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3.1
Radian Measure
Radian Measure ▪ Converting Between Degrees and Radians ▪
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3.1
Converting Degrees and Radians (page 94)
To Convert degree measure to radians.

• Multiply degree measure by 1 8 0
To Convert each radians measure to degrees.
• Multiply radian measure by
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180

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3.1
Example 1 Converting Degrees to Radians (page 94)
Convert each degree measure to radians.
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3.1
Example 2 Converting Radians to Degrees (page 94)
Convert each radian measure to degrees.
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1.3 Trigonometric Functions
Trigonometric Functions ▪ Right-Triangle-Based Definitions of
the Trigonometric Functions (Sec. 2.1) ▪ Quadrantal Angles
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2.1
Example: Finding Trigonometric Function Values of An
Acute Angle (page 46 – Cover with section 1.3)
Find the sine, cosine, and
tangent values for angles D
and E in the figure.
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2.1
Example: Finding Trigonometric Function Values of An
Acute Angle (cont.)
Find the sine, cosine, and
tangent values for angles D
and E in the figure.
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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 4(a) Finding Function Values of Quadrantal
Angles (page 24)
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 24)
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 (skip unitl chapter 5) ▪ Quotient Identities
(skip until chapter 5)
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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 = –.999
(b) cos θ = –1.7 (c) csc θ = 0
(a) cot θ = –.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
Extra Example Finding All Function Values Given One
Value and Condition (page 37 #77)
Find the five remaining trig function values for θ
given sec θ = -4, given that sin θ > 0.
Since sin is positive in quadrants I & II and sec is
negative in quadrants II & III we restrict our
discussion to quadrant II so , r = 4 and x = - 1.
4  (  1)  y so y 
2
sin  
2
2
y
15
r

15
4
The remaining functions follow
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2.1
Trigonometric Functions of Acute
Angles
Right-Triangle-Based Definitions of the Trigonometric Functions
(covered with section 1.3) ▪ Cofunction Identities (skip until
chapter 3) ▪ Trigonometric Function Values of Special Angles
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2.1
Example 4 Comparing Function Values of Acute Angles
(page 49)
Determine whether each statement is true or false.
(a) tan 25° < tan 23°
In the interval from 0° to 90°, as the angle increases,
the tangent of the angle increases.
tan 25° < tan 23° is false.
(b) csc 44° < csc 40°
In the interval from 0° to 90°, as the angle increases,
the sine of the angle increases, so the cosecant of
the angle decreases.
csc 44° < csc 40° is true.
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2.2
Trigonometric Functions of Non-Acute
Angles
Reference Angles ▪ Special Angles as Reference Angles ▪
Finding Angle Measures with Special Angles
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2.2
Example 1(a) Finding Reference Angles (page 55)
Find the reference angle for 294°.
294 ° lies in quadrant IV.
The reference angle is
360° – 294° = 66°.
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2.2
Example 1(b) Finding Reference Angles (page 55)
Find the reference angle for 883°.
Find a coterminal angle between 0° and 360° by
dividing 883° by 360°. The quotient is about 2.5.
883° is coterminal with 163°.
The reference angle is
180° – 163° = 17°.
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2.2
Example 2 Finding Trigonometric Functions of a
Quadrant II Angle (page 56)
Find the values of the six trigonometric functions for
135°.
The reference angle for 135° is 45°.
Choose point P on the terminal
side of the angle. The coordinates
of P are (1, –1).
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2.2
Example 2 Finding Trigonometric Functions of a
Quadrant II Angle (page 56)
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2.2
Example 3(a) Finding Trigonometric Function Values
Using Reference Angles (page 57)
Find the exact value of sin(–150°).
An angle of –150° is coterminal with an angle of
–150° + 360° = 210°.
The reference angle is
210° – 180° = 30°.
Since an angle of –150° lies in
quadrant III, its sine is negative.
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2.2
Example 3(b) Finding Trigonometric Function Values
Using Reference Angles (page 57)
Find the exact value of cot(780°).
An angle of 780° is coterminal with an angle of
780° – 2 ∙ 360° = 60°.
The reference angle is 60°.
Since an angle of 780°
lies in quadrant I, its
cotangent is positive.
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2.2
Example 4 Evaluating an Expression with Function
Values of Special Angles (page 57)
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3.1
Radian Measure (Part II)
Finding Function Values for Angles in Radians
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3.1
Example 3 Finding Function Values of Angles in Radian
Measure (page 97)
Find each function value.
1
 7 
o
o
(d) sin  
  sin (  2 1 0 )  sin (1 5 0 ) 
6 
2

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3.3
The Unit Circle and Circular Functions
Circular Functions ▪ Finding Values of Circular Functions ▪
Determining a Number with a Given Circular Function Value ▪
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3.3
Example 1 Finding Exact Circular Function Values
(page 113)
Find the exact values of sin (–3π), cos (–3π), and
tan (–3π).
An angle of –3π intersects the
unit circle at (–1, 0).
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3.3
Example 2(a) Finding Exact Circular Function Values
(page 113)
Use the figure to find
the exact values of
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3.3
Example 2(b) Finding Exact Circular Function Values
(page 113)
Use the figure and the
definition of tangent to
find the exact value of
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3.3
Example 2(b) Finding Exact Circular Function Values
(page 113)
Moving around the unit
circle
units in the
negative direction
yields the same
ending point as
moving around the
circle
units in the
positive direction.
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3.3
Example 2(b) Finding Exact Circular Function Values
(page 113)
corresponds to
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3.3
Example 2(c) Finding Exact Circular Function Values
(page 113)
Use reference angles and degree/radian conversion to
find the exact value of
In standard position, 330° lies in quadrant IV with a
reference angle of 30°, so
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3.3
Example 4(b) Finding a Number Given Its Circular
Function Value (page 114)
Approximate the value of s in the interval
Recall that
negative.
if
and in quadrant IV, tan s is
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3.2
Applications of Radian Measure
Arc Length on a Circle ▪ Area of a Sector of a Circle
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3.2
Example 1 Finding Arc Length Using s = rθ (page 101)
A circle has radius 25.60 cm. Find the length of the arc
intercepted by a central angle having each of the following
measures.
Convert θ to radians.
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3.2
Example 4 Finding an Angle Measure Using s = rθ
(page 102)
Two gears are adjusted so that the smaller gear drives the
larger one. If the radii of the gears are 3.6 in. and 5.4 in.,
and the smaller gear rotates through 150°, through how
many degrees will the larger gear rotate?
First find the radian measure of
the angle, and then find the arc
length on the smaller gear that
determines the motion of the
larger gear.
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3.2
Example 4 Finding an Angle Measure Using s = rθ
(cont.)
The arc length on the smaller gear is
An arc with length 3π cm on the larger gear corresponds
to an angle measure θ radians, where
The larger gear will rotate through 100°.
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3.2
Example 5 Finding the Area of a Sector (page 103)
Find the area of a sector of a circle having radius 15.20 ft
and central angle 108.0°.
The area of the sector is about 217.8 sq ft.
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3.4
Linear and Angular Speed
Linear Speed ▪ Angular Speed
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3.4
Example 1 Using Linear and Angular Speed Formulas
(page 122)
Suppose that P is on a circle with radius 15 in., and
ray OP is rotating with angular speed
radian per
second.
(a) Find the angle generated by P in 10 seconds.
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3.4
Example 1 Finding Exact Circular Function Values
(cont.)
(b) Find the distance traveled by P along the circle in
10 seconds.
from part (a)
(c) Find the linear speed of P in inches per second.
from part (b)
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