07/09/2023
Angles & Trigonometric
Functions
LESSON 1
1
Introduction
The term trigonometry comes from two Greek words –
trigonon, meaning triangle and metron, meaning measure.
Thus, trigonometry originally was a branch of mathematics
concerned with the measurement of the angles and sides
of a triangle.
Now the definition has been extended to include more
applications.
2
Definition
Trigonometry is that branch of
mathematics which deals with
the properties and applications
of particular ratios associated
with angles known as circular or
trigonometric functions.
3
1
07/09/2023
Angles
In geometry, we learned that angles are geometric figures formed
when two half-lines (rays) meet at one common point.
B
O
A
The two rays 𝑂𝐴 and 𝑂𝐵 are the sides of ∠𝐴𝑂𝐵 the common point O
is the vertex.
4
We shall now extend this concept of
an angle.
negative angle
Suppose we let a line segment rotate
around a fixed point.
The angle is the figure formed or
generated by the rotation of the line
segment around a fixed point.
The first position is called the initial side,
while the final position is called the
terminal side.
positive angle
5
Counter-clockwise direction
Clockwise direction
B
B
positive angle
O
O
A
negative angle
A
Greek letters and numbers may also be used to denote angles such as ∠𝜃, ∠𝛽, or ∠1, ∠2
6
2
07/09/2023
Angle Measurement
An angle is measured by
means of a protractor.
There are three standard
systems of measuring
angles:
▪
revolution system
▪
degree system
▪
radian system
7
Angle Measurement
In the revolution system, an
angle is measured by the
number of revolutions or
fraction of revolution made by
the terminal side in relation to
the initial side.
Example:
Unit of Measurement: rev
(equivalent to one whole angle)
8
Angle Measurement
The following is the relationship
between the degree system
and the revolution system:
360° = 1 𝑟𝑒𝑣
1
𝑟𝑒𝑣
2
1
90° = 𝑟𝑒𝑣
4
180° =
The degree is further divided into
minutes
′ and minutes into
seconds ′′
1° = 60′
1′ = 60′′
An angle may then measure as
20°15′ 30"
1
1° = 360 𝑟𝑒𝑣
9
3
07/09/2023
Angle Measurement
Scientific calculators with ° ′ "
key that can easily convert degreesminutes-seconds to decimals, and vice versa. Make sure your
calculator is set in DEGREE Mode.
Example:
1. Express 51°45′ 27"
51
°′"
45
°′"
27
°′"
Shift
=
° ′ " 51.7575°
2. Express 42.735° in degrees, minutes, and seconds
42.735
=
Shift
°′"
42°44′ 6"
10
Angle Measurement
Relationship between the degree
system, revolution system and radian
system.
The radian system is based on
the relation between the radius
and the circumference of a
circle.
1 rad =
1
2𝜋
rev or 0.159 rev
2𝜋 rad or 6.283 rad = 1 rev
2𝜋 rad = 360°
Unit of Measurement: radian (rad)
𝜋 rad = 180°
1rad =
1° =
180 °
𝜋
𝜋
180
or 57.296°
rad or 0.01745 rad
11
Angle Measurement
Example:
1. Express the following angles in 𝜋 radian:
30° = 30°
𝜋
𝑟𝑎𝑑
180
= 6 rad
𝜋
45° = 45°
𝜋
𝑟𝑎𝑑
180
=
𝜋
4
rad
60° = 60°
𝜋
𝑟𝑎𝑑
180
=
𝜋
3
rad
12
4
07/09/2023
Angle Measurement
Example:
2. Express 124° 30′ to radian
0.01745
𝑟𝑎𝑑
1
124.5°
3. Express
7𝜋
5
= 2.1725 rad
rad in degrees
7𝜋
180°
rad 𝜋 𝑟𝑎𝑑
5
=
(7)(180°)
5
= 252°
4. Express 3.5 radian in degrees, minutes, and seconds.
57.296°
1 𝑟𝑎𝑑
3.5 rad
= 200.536° = 200 °32′ 9.6"
13
Angle Measurement
Example:
5. Express
5𝜋
6
5𝜋
6
𝑟𝑎𝑑
radian in revolutions
1 𝑟𝑒𝑣
2𝜋 𝑟𝑎𝑑
5
= 12 rev
3
6. Convert rev to radians
8
3
2𝜋 𝑟𝑎𝑑
r𝑒𝑣 1 𝑟𝑒𝑣
8
=
3𝜋
rad
4
7. Express 5.23 radian in revolutions.
5.23 rad
0.159 𝑟𝑒𝑣
1 𝑟𝑎𝑑
= 0.8316 rev
14
5