CHAPTER 1.1
CHAPTER 1 TRIGONOMETRY
PART 1 – Radian and Degree Measure
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 1.0 – Students understand the notion of angle and how to measure it, in both
degrees and radians. They can convert between degrees and radians.
 19.0 - Students are adept at using trigonometry in a variety of applications
and word problems.
OBJECTIVE(S):
 Students will learn the definition of a radian.
 Students will learn how to sketch and find co terminal angles.
 Students will learn how to find complementary and supplementary
angles.
 Students will learn how to convert angles between degrees and
radians.
 Students will learn how to calculate an arc length.
 Students will learn how to calculate linear and angular speeds.
 Students will learn how to calculate the area of a sector.
Angles
As derived from the Greek language, the word trigonometry means
“____________________ ____ ___________________.” Initially, trigonometry dealt
with relationships among the sides and angles of triangles and was used in the
development of astronomy, navigation, and surveying. With the development of calculus
and the physical sciences in the 17th century, a different perspective arose – one that
viewed the classic trigonometric relationships as functions with the set of real numbers as
their domains. Consequently, the applications of trigonometry expanded to include a vast
number of physical phenomena involving rotations and vibrations. These phenomena
include sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits
of atomic particles.
The approach in this text incorporates both perspectives, starting with angles and their
measure.
An ______________ is determined by rotating a ray (half-line) about its endpoint. The
starting position of the ray is the __________________ ___________ of the angle, and
the position after rotation is the ___________________ __________. The endpoint of
the ray is the _____________ of the angle.
Vertex
Angle
CHAPTER 1.1
This perception of an angle fits a coordinate system in which the origin is the vertex and
the initial side coincides with the positive x-axis. Such an angle is in
___________________ ___________________.
Terminal
side
y
Initial side
x
Angle in Standard Position
____________________ ____________ are generated by counterclockwise rotation, and
________________ ___________ by clockwise rotation.
y
Positive angle
(counterclockwise)
x
Negative angle
(clockwise)
Angles are labeled with Greek letters _____ (alpha), _____ (beta), and _____ (theta), as
well as uppercase letters A, B, and C.
y
y
x
x
CHAPTER 1.1
Radian Measure
The measure of an angle is determined by the amount of rotation from the initial side to
the terminal side. One way to measure angles is in ______________. This type of
measure is especially useful in calculus. To define a radian, you can use a central angle
of a circle, one whose vertex is the center of the circle.
y
x
Arc length (s) = radius (r) when  = ___ radian
Definition of Radian
One radian is the measure of a central angle  that intercepts an arc s equal in length to
the radius r of the circle. In other words the following relationship is true:
 = _____
where  is measured in radians.
Because the circumference of a circle is __________ units, it follows that a central angle
of one full revolution (counterclockwise) corresponds to an arc length of
s = ____________
One revolution around a circle of radius r corresponds to an angle of ____ radians
because
=
=
=
radians
CHAPTER 1.1
Moreover, because 2  __________, there are just over six radius lengths in a full
circle.
y
x
1.) Estimate the angle to the nearest one-half radian.
a.
b.
c.
Because the units of measure for s and r are the same, the ratio_______ has no units – it
is simply a real number.
CHAPTER 1.1
Because the radian measure of an angle of one full revolution is ______, you can obtain
the following:
1
revolution =
2
=
radians
1
revolution =
4
=
radians
1
revolution =
6
=
radians

6

4

3

2

2
CHAPTER 1.1
Recall that the four quadrants in a coordinate system are numbered ____, ____, ____, and
____.
 = ____
Quadrant ____

Quadrant ____

 = ____
 = ____
Quadrant ____

Quadrant ____

 = _____
2.) Determine the quadrant in which each angle lies.
11
a.
b. -1
8
3.) Sketch each angle in standard position.
7
a. 
4
Note that angles between 0 and
 are _______________ angles.
DAY 1
c. -4.25
b. 4


are ______________ angles and angles between and
2
2
CHAPTER 1.1
Two angles are co terminal if they have the same __________ and ______________
sides. For instance,
the angles _____ and ______ are co terminal, as are the angles _____ and _______. You
can find an angle that is co terminal to a given angle  by adding or subtracting _______
(one revolution). A given angle  has infinitely many co terminal angles. For instance,

  is co terminal with _________________ where n is an integer.
6
EXAMPLE 1: Sketching and Finding Co terminal Angles
13
a.) For the positive angle
, subtract _______ to obtain a co terminal angle.
6
13
 2
6
=
=

2

0
3
2
CHAPTER 1.1
b.) For the positive angle
3
 2
4
3
, subtract _______ to obtain a co terminal angle.
4
=
=

2

0
3
2
CHAPTER 1.1
c.) For the negative angle 

2
 2
3
2
, add _______ to obtain a co terminal angle.
3
=
=

2

0
3
2
4.) Determine two co terminal angles (one positive and one negative) for each angle.
Give your answer in radians.
11
9

a.  
b.  
c.   
6
4
3
CHAPTER 1.1
Two positive angles  and  are complementary (complements of each other) if their
sum is ______. Two positive angles are supplementary (supplements of each other) if
their sum is ____.




Complementary Angles
Supplementary Angles
EXAMPLE 2: Complementary and Supplementary Angles
2
4
If possible, find the complement and the supplement of a.)
and b.)
.
5
5
2
a.) The complement of
is
5
 2

=
=
2 5
The supplement of

2
=
5
b.) The complement of

2

4
=
5
2
is
5
=
4
is
5
=
The complement ___________ _______ ____________ because complements
must be _______________ angles.
The supplement of

4
=
5
4
is
5
=
CHAPTER 1.1
5.) Find, if possible, the complement and supplement of each angle.
2
a.
b. 2.5
7
Degree Measure
A second way to measure angles is in terms of degrees, denoted by the symbol ____. A
measure of one degree (____) is equivalent to a rotation of ______ of a complete
revolution about the vertex. To measure angles, it is convenient to mark degrees on the
circumference of a circle.
y
x
So, a full revolution (counterclockwise) corresponds to _________, a half revolution to
_______, a quarter revolution to ______, and so on.
CHAPTER 1.1
Because 2 radians corresponds to _____ complete revolution, degrees and radians are
related by the equations
3600 = _____ rad
and
180 0 = _____ rad
From the latter equation, you obtain
10 = _____ rad
and
1 rad = _____
Conversions Between Degrees and Radians
1.
To convert degrees to radians, multiply degrees by _________.
2.
To convert radians to degrees, multiply radians by __________ (or simply
substitute 180 0 for  when radians are written in terms of  .)

= _____
6

= _____
4

= _____
3

= _____
2
 = ______
2 = _____
When no units of angle measure are specified, radian measure is implied. For instance, if
you write  = 2, you imply that  = _____________.
CHAPTER 1.1
EXAMPLE 3: Converting from Degrees to Radians
a.) 1350
b.) 540 0
=
Multiply by __
=
=
=
=
=
Multiply by __
c.)  270 0
=
Multiply by __
=
=
6.) Rewrite each angle in radian measure as a multiple of  (Do not use a calculator).
a. 210 0
b.  315 0
c. 150 0
CHAPTER 1.1
EXAMPLE 4: Converting from Radians to Degrees

9
a.)  rad
b.)
rad
2
2
=
Multiply by __
=
=
Substitute
______ for  .
=
=
c.) 2 rad
=
Multiply by __
=

DAY 2
With calculators it is convenient to use decimal degrees to denote fractional parts of
degrees. Historically, however, fractional parts of degrees were expressed in minutes and
seconds, using the prime (____) and double prime (____) notations, respectively. That is,
1' = ________________________ = _____________
1" = ________________________ = _____________
Consequently, an angle of 64 degrees, 32 minutes, and 47 seconds is represented by  =
____________________.
CHAPTER 1.1
114.59 0 to minutes and seconds:
114 degrees + .59(60 minutes)
7.) Rewrite each angle in degree measure (Decimal form then in terms of degrees,
minutes and seconds).

11
a.
b. -3
c.
4
8
8.) Convert each angle measure to decimal degree form.
a. 610 17' 5"
b. 80 0 25' 50"
c.  135 0 40'
CHAPTER 1.1
Applications
The radian measure formula, ____________, can be used to measure arc length along a
circle.
Arc Length
For a circle of radius r, a central angle  intercepts an arc of length s given by
Length of circular arc
where  is measured in radians. Note that if r = 1, then s =  , and the radian measure of
 equals the arc length.
EXAMPLE 5: Finding Arc Length
A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle
of 240 0 .
To use the formula _____________, first convert 240 0 to radian measure.
240 0 =
=
=
radians
Then, using a radius of r = _____inches, you can find the arc length to be
s  r =
=

inches
CHAPTER 1.1
Note that the units for r are determined by the units for r because  is given in radian
measure, which has no units.
9.) A circle has a radius of 12 inches. If the central angle is 150 0 , find the length of the
arc intercepted by the central angle.
10.) Find the central angle if the arc length of a circle is 17 cm and the radius is 8 cm.
DAY 3
The formula for the length of a circular arc can be used to analyze the motion of a particle
moving at a constant speed along a circular path.
Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular arc of radius r. If s is the
length of the arc traveled in time t, then the linear speed v of the particle is
Linear speed v =
=
CHAPTER 1.1
Moreover, if  is the angle (in radian measure) corresponding to the arc length s, then the
angular speed  (the lowercase Greek letter omega) of the particle is
Angular speed  =
=
EXAMPLE 6: Finding Linear Speed
The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of
this second hand as it passes around the clock face.
In one revolution, the arc length traveled is
s
=
2r
=
Substitute for r.
=
The time required for the second hand to travel this distance is t = 1 minute = 60 seconds.
So, the linear speed of the tip of the second hand is
s
Linear speed =
t
=

EXAMPLE 7: Finding Angular and Linear Speeds
A Ferris wheel with a 50-foot radius makes 1.5 revolutions per minute.
a.) Find the angular speed of the Ferris wheel in radians per minute.
Because each revolution generates ____ radians, it follows that the wheel turns
1.52 = _____ radians per minute. In other words, the angular speed is
Angular speed =
=
=
CHAPTER 1.1
b.) The linear speed is
Linear speed =
=

=
1
11.) An 8 -inch circular power saw rotates at 4600 revolutions per minute.
2
a. Find the angular speed of the saw blade in radians per minute.
b. Find the linear speed (in feet per minute) of one of the 24 cutting teeth as they
contact the wood being cut.
A sector of a circle is the region bounded by two radii of the circle and their intercepted
arc.
CHAPTER 1.1
Area of a Sector of a Circle
For a circle of radius r, the area A of a sector of the circle with central angle  is given
by
A=
where  is measured in radians.
EXAMPLE 8: Area of a Sector of a Circle
A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and
rotates through an angle of 120 0 . Find the area of the fairway watered by the sprinkler.
First convert 120 0 to radian measure as follows.

=
120 0
=
Multiply by ___________.
=
Then, using  = _______ and r = ______, the area is
A
=
1 2
r 
2
Formula for the area of a sector of a
circle.
=
Substitute for r and  .
=
Simplify.
=
Simplify.
CHAPTER 1.1
12.) A sprinkler system on a farm is set to spray water over a distance of 45 meters and
to rotate through an angle of 1350 . Find the area of the region.
DAY 4