Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved....
-
Upload
ginger-collins -
Category
Documents
-
view
215 -
download
0
Transcript of Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved....
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Chapter 3Chapter 3
Radian Measure and Radian Measure and the Unit Circle the Unit Circle
ApproachApproach
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Chapter 3Chapter 3OverviewOverview
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Chapter 3Chapter 3ObjectivesObjectives
Convert between degrees and radians. Calculate arc length and the area of a circular sector. Relate angular and linear speeds. Draw the unit circle and label the sine and cosine values
for special angles (in both degree and radians.)
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Section 3.1Section 3.1Radian MeasureRadian Measure
Skills Objectives Calculate the radian
measure of an angle. Convert between degrees
and radians. Calculate trigonometric
function values for angles given in radians.
Conceptual Objectives Understand that degrees
and radians are both measures of angles.
Realize that radian measure allows us to write trigonometric functions as functions of real numbers.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Radian MeasureRadian Measure
To correctly calculate radians from the
formula , the radius and arc length
must be expressed in the same units.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Your Turn: Calculate Radian MeasureYour Turn: Calculate Radian Measure
•What is the measure (in radians) of a central
angle that intercepts an arc of length 12 mm
on a circle with radius 4 cm?
•Solution: First, convert 4 cm to 40 mm.
•Next, calculate = 0.3.40
12
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Converting Between Degrees and RadiansConverting Between Degrees and Radians
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Your Turn : Converting Between Degrees Your Turn : Converting Between Degrees and Radiansand Radians
Convert 60° to radians.
Solution: r = 60 = or 1.047.
Convert to radians.
Solution: d = = 270°.
180
3
180
2
3
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Special Angles in the Unit CircleSpecial Angles in the Unit Circle
We can now draw the unit circle with special angles in degrees and radians.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Section 3.2 Section 3.2 Arc Length and Area of a Circular Arc Length and Area of a Circular
SectionSection
Skills Objectives Calculate the length of an
arc along a circle. Find the area of a circular
sector. Solve application
problems involving circular arc lengths and sectors.
Conceptual Objectives Understand that to use
the arc length formula, the angle measure must be in radians.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Arc LengthArc Length
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Your Turn: Calculating Arc LengthYour Turn: Calculating Arc Length
In a circle with radius 15 inches, an arc is intercepted by a central angle with measure . Find the arc length.
Solution: s = 15 = 5inches.
In a circle with radius 20 m, an arc is intercepted by a central angle with measure 113°. Find the arc length. Approximate the arc length to the nearest meter.
Solution: s = ≈ 39 m
3
3
18011320
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Find the area of a slice of pizza (cut into 8 equal pieces) if the entire pizza has a 16-inch diameter.
Solution: r = 8, r = , so area = 25 in.2 .
Calculating Area of a Circular SectorCalculating Area of a Circular Sector
4
28
42
1
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Section 3.3 Section 3.3 Linear and Angular SpeedsLinear and Angular Speeds
Skills Objectives Calculate linear
speed. Calculate angular
speed. Solve application
problems involving angular and linear speeds.
Conceptual Objectives Relate angular speed to
linear speed.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Linear and Angular SpeedLinear and Angular Speed
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Your Turn: Linear SpeedYour Turn: Linear Speed
A car travels at a constant speed around a circular track with circumference equal to 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed in miles per hour?
Solution: t = 12 minutes, or hour and s = 7 X 3.
V = = = 105 mph.
5
1
t
s2.
21
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Relationship Between Linear and Relationship Between Linear and Angular SpeedsAngular Speeds
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Section 3.4 Section 3.4 Definition 3 of Trigonometric Functions: Definition 3 of Trigonometric Functions:
Unit Circle ApproachUnit Circle Approach
Skills Objectives Draw the unit circle
illustrating the special angles and label the sine and cosine values.
Determine the domain and range of trigonometric (circular) functions.
Classify circular functions as even or odd.
Conceptual Objectives Understand that trigonometric
functions using the unit circle approach are consistent with both of the previous definitions (right triangle trigonometry and trigonometric functions of nonacute angles in the Cartesian plane).
Relate x-coordinates and y-coordinates of points on the unit circle to the values of cosine and sine functions.
Visualize periodic properties of trigonometric (circular) functions.
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
The Unit CircleThe Unit Circle
Tri
go
no
met
ry, T
hir
d E
dit
ion
by
Cyn
thia
Y.
Yo
un
g, ©
201
2 Jo
hn
Wile
y an
d S
on
s. A
ll ri
gh
ts
rese
rved
.
Common Mistake: Using the Wrong Mode Common Mistake: Using the Wrong Mode on a Calculatoron a Calculator
Many calculators automatically reset to degree mode after every calculation, so be sure to always check what mode the calculator indicates.