Chapter 5
Trigonometric Functions
Section 5.1
Angles and Arcs
Definition of an Angle
• An angle is formed by rotating a given ray
about its endpoint to some terminal position.
• The original ray is the initial side of the angle,
and the second ray is the terminal side of the
angle.
• The common endpoint is the vertex of the
angle.
Definition of an Angle
• There are several methods that can be used to
name an angle. Figure 5.2 below shows and
angle that is represented using the Greek
letter, a, and is designated as a.
• It could also be named
O,  AOB, or  BOA.
It is traditional to list the
vertex between the
other two points.
Definition of an Angle
• Angles formed by a counterclockwise rotation
are considered positive angles.
• Angles formed by a clockwise rotation are
considered negative angles.
Degree Measure
• The measure of an angle is determined by the
amount of rotation of the initial ray.
– The concept of measuring angles in degrees grew
out of the belief of the early Sumerians and
Babylonians that the seasons repeated every 360
days.
Definition of Degree
• One degree is the measure of an angle formed
by rotating a ray 1/360 of a complete
revolution.
• The symbol for degree is 0.
Classification of Angles
• 1800 angles are straight angles.
• 900 angles are right angles.
• Angles that have a measure greater than 00
but less than 900 are acute angles.
• Angles that have a measure greater than 900
but less than 1800 are obtuse angles.
Classification of Angles
• An angle superimposed in a Cartesian
coordinate system is in standard position if its
vertex is at the origin and its initial side is on
the positive x-axis.
Classification of Angles
• Two positive angles are complementary angles
if the sum of the measures of the angles is
900.
– Each angle is the complement of the other angle.
Classification of Angles
• Two positive angles are supplementary angles
if the sum of the measures of the angles is
1800.
– Each angle is the supplement of the other angle.
Example 1
For each angle, find the measure (if possible) of
its complement and of its supplement.
a. q = 400
b. q = 1250
Classification of Angles
Are the two acute angles of any right triangle
complementary angles? Explain.
Yes. The sum of the measures of the angles of
any triangle is 1800. The right angle has a
measure of 900. Thus the measure of the sum of
the two acute angles must be 1800 – 900 = 900.
Classification of Angles
• It is possible for an angle to have a measure
that is greater than 3600. In this scenario an
angle was formed by rotating its terminal side
more than one revolution of the initial ray.
Classification of Angles
• If the terminal side of an angle in standard
position lies on a coordinate axis, then the
angle is classified as a quadrantal angle.
– The 900 angle, the 1800 angle, and the 2700 angle
are all examples of quadrantal angles.
Classification of Angles
• Angles in standard position that have the
same terminal sides are co-terminal angles.
• Every angle has an unlimited number of coterminal angles.
Classification of Angles
Given  q in standard position with measure x0,
then the measures of the angles that are coterminal with  q are given by:
x0 + k  3600
where k is an integer.
Example 2
Assume the following angles are in standard
position. Classify each angle by quadrant, and
then determine the measure of the positive
angle with measure less than 3600 that is coterminal with the given angle.
a = 5500
b = -2250
g = 11050
Conversion Between Units
• There are two popular methods for
representing a fractional part of a degree.
• Decimal Degrees
– 29.760
• DMS (Degree, Minute, Second)
– 1260 12’ 27”
Conversion Between Units
• In the DMS method, a degree is subdivided
into 60 equal parts, each of which is called a
minute, denoted by ‘. Therefore 10 = 60’
Furthermore, a minute is divided into 60
equal parts, each of which is called a second,
denoted by “. Thus 1’ = 60” and 10 = 3600”.
10/60’ = 1
1’/60” = 1
10/3600” = 1
Example
Convert 1260 12’ 27” to decimal degrees.
Example
Convert 31.570 to DMS.
Radian Measure
Another commonly used angle measurement is
the radian. To define a radian, first consider a
circle of radius, r, and two radii OA and OB. The
angle, q, formed by the two radii is a central
angle. The portion of the circle between A and B
is an arc. We say that arc AB subtends the angle
q. The length of arc AB is s.
Definition of Radian
• One radian is the measure of the central angle
subtended by an arc of length r on a circle of
radius r.
• Given an arc of length s on a circle of radius r,
the measure of the central angle subtended
by the are is q = s/r radians.
Radian Measure
As an example, consider than an arc of length 15
centimeters on a circle with a radius of 5
centimeters subtends and angle of 3 radians, as
shown in Figure 5.22. The same result can be
found by 15 centimeters by 5 centimeters.
Radian-Degree Conversion
• To convert from radians to degrees, multiply
by 1800/p.
• To convert from degrees to radians, multiply
by p/1800.
Radian-Degree Conversion
Example 3
Convert 3000 to radians.
Example 4
−3
Convert p
4
radians to degrees.
Arcs and Arc Length
• Let r be the length of the radius of a circle and
q the nonnegative radian measure of a central
angle of the circle. Then the length of the arc,
s, that subtends the central angle is s = rq.
Example 5
Find the length of an arc that subtends a central
angle of 1200 in a circle of radius 10 centimeters.
Example 6
A pulley with a radius of 10 inches uses a belt to
drive a pulley with a radius of 6 inches. Find the
angle through which the smaller pulley turns as
the 10-inch pulley makes on revolution. State
your answer in radians and also degrees.
Linear and Angular Speed
• Linear speed, n, is distance traveled per unit
time.
• Angular speed, w, is the angle through which a
point on the circle moves per unit time.
Linear and Angular Speed
• Linear Speed 
n=
𝑠
𝑡
– Where n is the linear speed, s is the distance
traveled, and t is the time.
• Angular Speed
 w=
θ
𝑡
– Where w is the angular speed, q is the measure (in
radians) of the angle through which a point has
moved, and t is the time.
Example 7
A hard disk in a computer rotates at 3600 revolutions
per minute. Find the angular speed of the disk in
radians per second.
3600 rev/minute = 3600 rev ∙ 2p radians ∙ 1 minute
1 minute
1 rev
60 sec
= 120p radians/second
≈ 377 radians/second
Linear and Angular Speed
A wheel has both linear and angular speed. As
the wheel moves a distance, s, point A moves
through an angle, q. the arc length subtending
angle q is also s, the distance traveled by the
wheel.
Linear and Angular Speed
The formula from the previous slide, u = rw
gives the linear speed of a point on a rotating
body in terms of distance r from the axis of
rotation and the angular speed w, provided that
w is in radians per unit of time.
Example 8
A wind machine is used to generate electricity.
The wind machine has propeller blades that are
12 feet in length. If the propeller is rotating at 3
revolutions per second, what is the linear speed,
in feet per second, of the tips of the blades?
Assignments
Day #1
Pgs. 472-473 (1-57 odd)
Day #2
Pg. 473 (59-79 odd)