Transcript Lecture Note PowerPoint Slides

Radian Measure and
the Unit Circle
Approach
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Chapter 3
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Chapter 3
Overview




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.)
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Chapter 3
Objectives
Skills Objectives
Conceptual Objectives
 Understand that degrees
 Calculate the radian
and radians are both
measure of an angle.
measures of angles.
 Convert between degrees
 Realize that radian
and radians.
measure allows us to write
 Calculate trigonometric
trigonometric functions as
function values for angles
functions of real numbers.
given in radians.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Section 3.1
Radian Measure
To correctly calculate radians from the
s
formula q = , the radius and arc length
r
must be expressed in the same units.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
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  = 12  0.3.
40
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Your Turn: Calculate Radian Measure
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Converting Between Degrees and Radians
 Convert 60° to radians.


 Solution: r = 60
=
or 1.047.
180
3
s
 Convert q = to radians.
r
3 180

 Solution: d =
= 270°.
2

Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Your Turn : Converting Between Degrees
and Radians
We can now draw the unit circle with special
angles in degrees and radians.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Special Angles in the Unit Circle
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.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Section 3.2
Arc Length and Area of a Circular
Section
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Arc Length
 In a circle with radius 15 inches, an arc is

intercepted by a central angle with measure
.
3
Find the arc length.
 
 Solution: s = 15  3  = 5 inches.
 
 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 = 20  113   ≈ 39 m
180
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Your Turn: Calculating Arc Length
 Find the area of a slice of pizza (cut into 8 equal pieces)
if the entire pizza has a 16-inch diameter.

1  2
 Solution: r = 8, r =
, so area =   8  25 in.2 .
4
2 4
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Calculating Area of a Circular Sector
Skills Objectives
 Calculate linear
speed.
 Calculate angular
speed.
 Solve application
problems involving
angular and linear
speeds.
Conceptual Objectives
 Relate angular speed to
linear speed.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Section 3.3
Linear and Angular Speeds
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Linear and Angular 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?
1
Solution: t = 12 minutes, or
hour and
5
s = 7 X 3.
s
21
V =
=
= 105 mph.
.2
t
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Your Turn: Linear Speed
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Relationship Between Linear and
Angular Speeds
Skills Objectives
Conceptual Objectives
 Understand that trigonometric
 Draw the unit circle
functions using the unit circle
illustrating the special
approach are consistent with both
angles and label the sine
of the previous definitions (right
and cosine values.
triangle trigonometry and
trigonometric functions of
 Determine the domain
nonacute angles in the Cartesian
and range of trigonometric
plane).
(circular) functions.
 Relate x-coordinates and y Classify circular functions
coordinates of points on the unit
circle to the values of cosine and
as even or odd.
sine functions.
 Visualize periodic properties of
trigonometric (circular) functions.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Section 3.4
Definition 3 of Trigonometric Functions:
Unit Circle Approach
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
The Unit Circle
Many calculators automatically reset to degree
mode after every calculation, so be sure to
always check what mode the calculator
indicates.
Trigonometry, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights
Common Mistake: Using the Wrong Mode
on a Calculator