PRECALCULUS

ANGLES AND
THEIR MEASURE
Second Quarter – Week 01
Prepared by: JEROME R. PASCUAL
Lesson Objectives:
Define an angle.
Convert angle measures.
Illustrate positive and negative angles, angles in
standard position, quadrantal angles, coterminal
angles, and reference angle.
Introduction
Trigonometry, a branch of mathematics with ancient
origins, is a fundamental tool for understanding the
relationships between angles and sides in triangles. It
provides a framework for analyzing and solving a wide
range of problems, from measuring distances to
modeling oscillatory motion. Trigonometry's
applications span across various fields, including
physics, engineering, navigation, and even art, making it
an essential topic in mathematics education.
Introduction
Trigonometry is a branch of mathematics that deals with the study
of the relationships between the angles and sides of triangles. It
encompasses the use of trigonometric functions like sine, cosine,
and tangent to analyze and calculate various properties of triangles
and the periodic behavior of functions. One concise and clear
definition comes from the mathematician and philosopher Carl
Friedrich Gauss, who described trigonometry as "the science of
triangles and the relationships between their angles and sides."
This definition highlights the fundamental nature of trigonometry in
understanding geometric and periodic phenomena.
Definition of an Angle
An angle in geometry is defined as the
figure formed by two rays with a common
endpoint. One of the classic definitions of
an angle is attributed to the Greek
mathematician Euclid, who described it in
his work "Elements" as "the inclination to
one another of two lines in a plane which
meet one another and do not lie in a
straight line."
Parts of an Angle

Side - two noncollinear rays with common Vertex – common endpoint of two sides.
endpoint.
Initial Side – stationary side of an angle.
Vertex – common endpoint of two rays.
Terminal Side – rotating side of an angle.
Unit of Measures of an Angle
Angles can be measured using various units, and
the most common units of angle measurement are:

1. Degrees (°): Degrees are the most widely used

unit for measuring angles. A full circle is divided
into 360 degrees. Each degree can be further
subdivided into minutes (') and seconds ("). For
example, 1 degree is equal to 60 minutes, and 1
minute is equal to 60 seconds.
Unit of Measures of an Angle
Angles can be measured using various units, and
the most common units of angle measurement are:

2. Radians (rad): Radians are a unit of angular

measurement based on the radius of a circle. One radian is
defined as the angle subtended at the center of a circle
when the arc's length is equal to the circle's radius. There
are approximately 2π radians in a full circle, which is
equivalent to 360 degrees. Radians are commonly used in
advanced mathematics, physics, and engineering,
particularly in calculus and trigonometry.
Unit of Measures of an Angle
Angles can be measured using various units, and
the most common units of angle measurement are:

3. Revolutions (rev or rot): This unit measures angles

in terms of complete revolutions around a circle. One
complete revolution is equivalent to 360 degrees or 2π
radians. It is often used in engineering and machinery
where the focus is on rotational motion.
Unit of Measures of an Angle
Angles can be measured using various units, and
the most common units of angle measurement are:

4. Gradians (gon or grad): Gradians are another unit of

angular measurement, though they are less commonly
used. In this system, a full circle is divided into 400
gradians, making it a base - 10 system. A right angle is
equivalent to 100 gradians. Gradians are used in some
engineering and surveying applications, but they are
not as prevalent as degrees or radians.
Conversion of Angle Measures
Example:
𝟖𝝅
Convert 𝟑 rad to degree measure.

Convert 𝟏𝟓° to radian measure.

Conversion of Angle Measures
Practice:
Convert the following to degree measure
𝜋
1. rad
3
7𝜋
2. rad
3
17𝜋
3. rad
6
Convert the following to radian measure
1. 120°
2. 315°
3. 975°
Angle Rotation
Positive and negative angles refer to the
direction in which an angle is measured
relative to initial side. In a standard
coordinate system, typically in
mathematics and physics, angles are
measured counterclockwise from an initial
side as positive angles and clockwise as
negative angles.
Angle in Standard Position
An angle in standard position is a concept
in trigonometry that refers to the position
of an angle when its vertex is located at the
origin of a coordinate plane, and one of the
rays, known as the initial side, lies along
the positive x-axis. The other ray, called
the terminal side, is drawn from the initial
side to form the angle.
Angle in Standard Position
Example: Sketch 30° and −135° in standard position.
Angle in Standard Position
Practice: Sketch the following angles in standard position.

1. 60° 4. −215°
2. 120° 5.
7𝜋
6
3. −45° 7𝜋
6. −
4
Quadrantal Angles
A quadrantal angle is an angle in standard
position that terminates on one of the axes (x-axis
or y-axis).

Quadrantal angles in standard position include:

0°: This angle lies along the positive x-axis.

90°: This angle lies along the positive y-axis.
180°: This angle lies along the negative x-axis.
270°: This angle lies along the negative y-axis.
Angle Notation

In trigonometry, angles are often denoted

by Greek letters, such as θ (theta) or φ
(phi).

It's important to note that the context in

which the angle is used will often dictate
the specific notation that is employed.
Coterminal Angles
Coterminal angles are angles that share the same
initial and terminal sides but may differ in their
measures. In other words, two angles are coterminal if
they end in the same position when their initial side is
placed on the coordinate plane.

Formally, two angles, 𝜃 and 𝛼, are coterminal if there

exists an integer 𝑛 such that:
𝜃 = 𝛼 + 360𝑛 or α = 𝜃 + 360𝑛
𝜃 = 𝛼 + 2𝜋𝑛 or α = 𝜃 + 2𝜋𝑛
Coterminal Angles
Example: Give the largest negative and smallest positive coterminal angles with the following
angles SOLUTION (No. 1): SOLUTION (No. 2): SOLUTION (No. 3):
11𝜋
SOLUTION (No. 4):
𝜋
𝜃 = 80° 𝜃 = 100° 𝜃=−
𝜃= 3
𝑎 = 80 + 360𝑛 𝑎 = 100 + 360𝑛 9
1. 80° 𝑎=
11𝜋
+ 2𝜋𝑛
𝜋
𝑎 = − + 2𝜋𝑛
3
If 𝑛 = 1 If 𝑛 = 1 9
𝑎 = 80 + 360 = 440° 𝑎 = 100 + 360 = 460° If 𝑛 = 1
If 𝑛 = 1
2. 100° If 𝑛 = −1 If 𝑛 = −1 11𝜋 29𝜋 𝑎=−
𝜋
+ 2𝜋 =
5𝜋
𝑎= + 2𝜋 = 3 3
𝑎 = 80 − 360 = −280° 𝑎 = 100 − 360 = −260° 9 9
If 𝑛 = −1 If 𝑛 = −1
11𝜋 7𝜋 𝜋 7𝜋
11𝜋 Therefore, the smallest positive Therefore, the smallest positive 𝑎 = − − 2𝜋 = −
3. rad coterminal angle and the largest coterminal angle and the largest
𝑎=
9
− 2𝜋 = −
9 3 3
9 negative coterminal angle with negative coterminal angle with
Therefore, the smallest positive Therefore, the smallest positive
80° are 440° and −280° 100° are 460° and −260° and the largest negative
and the largest negative
respectively. respectively. 11𝜋 coterminal angle with − 3
𝜋
𝜋 coterminal angle with 9
4. − 3 rad are
29𝜋 7𝜋
and − 9 respectively. are
5𝜋
3
7𝜋
and − respectively.
3
9
Coterminal Angles
State if the given angles are coterminal:
1. 355° ∶ 555°
2. 100° ∶ −440°
3. 20° ∶ 220°
4. 55° ∶ −415°
5. 85° ∶ −275°
Reference Angles
The reference angle for any angle in
standard position is the positive acute
angle between the terminal side of the
angle and the x-axis.

Reference angle is always positive and

always between 0° and 90° . The
reference angle of a given angle 𝜃 will
be denoted as 𝜃 ′ (read as theta prime).
Reference Angles
Example: Find the reference angle of each of the following angles.

1. 30° 5. −60°

2. 135° 6. −170°

3. 240° 7. −230

4. 330° 8. −315°
THANK YOU!