Angles and their
Measures
Lesson 1

As derived from the Greek Language, the
word trigonometry means “measurement of
triangles.”

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 domain.

Consequently the applications expanded to
include physical phenomena involving rotations
and vibrations, including sound waves, light
rays, planetary orbits, vibrating strings,
pendulums, and orbits of atomic particles.

We will explore both perspectives beginning
with angles and their measures…..

An angle is determined by rotating a ray
about its endpoint.

The starting position of called the initial
side. The ending position is called the
terminal side.
Standard Position
Vertex is at the origin, and the initial side is on the x-axis.
90 
II
I
0  , 360 
180 
Initial Side
IV
III
270 

Positive Angles are generated by
counterclockwise rotation.

Negative Angles are generated by clockwise
rotation.

Let’s take a look at how negative angles are
labeled on the coordinate graph.
Negative Angles
Go in a Clockwise rotation
 270 
0  ,  360 
 180 
 45
 90 

Coterminal Angles

Angles that have the same initial and terminal
side. See the examples below.
Coterminal Angles
They have the same initial and terminal sides.
Determine 2 coterminal angles, one
positive and one negative for a 60
degree angle.
60 
60 + 360 = 420 degrees
60 – 360 = -300 degrees
Give 2 coterminal angles.
30
30 + 360 = 390 degrees
30 – 360 = -330 degrees
Give a coterminal angle, one
positive and one negative.
230
230 + 360 = 590 degrees
230 – 360 = -130 degrees
Give a coterminal angle, one
positive and one negative.
20
-20 + 360 = 340 degrees
-20 – 360 = -380 degrees
Give a coterminal angle, one
positive and one negative.
460

460 + 360 = 820 degrees
460 – 360 = 100 degrees
100 – 360 = -260 degrees
Good but
not best
answer.
Complementary Angles
Sum of the angles is 90
Find the complement of each angles:
40
40 + x = 90
x = 50 degrees
120
No Complement!
Supplementary Angles
Sum of the angles is 180
Find the supplement of each angles:
40
120
40 + x = 180
120 + x = 180
x = 140 degrees
x = 60 degrees
Coterminal Angles:
Angle  360
To find a Complementary Angle:
To find a Supplementary Angle:
90  Angle
180  Angle
Radian Measure


One radian is the measure of the central
angle,  , that intercepts an arc, s, that is
equal in length to the radius r of the circle.
C  2 r
C
2r

2
2
C
r
2

So…1 revolution is equal to 2π radians

2 radians  360
 radians  180


2
radians  90


Let’s take a
look at them
placed on the
unit circle.
Radians
Now, let’s add more…..
1.57 rad 
Radians
3
radians
4

4

1  
 
2 2 

4
radians

2
3.14 rad 
5
radians
4
6.28 rad 
7
radians
4
4.71 rad 
More Common Angles
Let’s take a look at more common angles that
are found in the unit circle.
Radians
3
radians
4

4
radians
3.14 rad 
5
radians
4
6.28 rad 
7
radians
4
Radians
3
radians
4

4
radians
3.14 rad 
5
radians
4
6.28 rad 
7
radians
4
Look at the Quadrants
Determine the Quadrant of the
terminal side of each given angle.

3
7
12
2

3
371
Q1
Go a little more than one quadrant – negative. Q3
A little more than one revolution. Q1
Determine the Quadrant of the
terminal side of each given angle.
14
5
156
9
8
240
1000
Q3
Q2
2 Rev + 280 degrees. Q4
Coterminal Angles using
Radians

Find a coterminal angle.

There are an infinite number of coterminal angles!
Give a coterminal angle, one
positive and one negative.
13
4
Give a coterminal angle, one
positive and one negative.

5
Find the complement of each
angles:
2
5
Find the supplement of each
angles:
2
5
Find the complement &
supplement of each angles, if
possible:
2
3
None
Coterminal Angles:
Angle  2
To find a Complementary Angle:

2
To find a Supplementary Angle:
RECAP
 Angle
  Angle
Conversions
1 


180
radians
1 radian 
180


135 3
135 x


180 180
4


NOTE: The answer is in radians!
270
3
 270 x


180
180
2


9 180

x
 810
2

Convert 2 radians to degrees
180 360

2x

 114.59


Arc Length

The relationship between arc length, radius,
and central angle is
Arc Length = (radius) (angle)
1st Change 240 degrees into radians.
240 4
240 x


radians
180
180
3


 4 
s  4

 3 
16
s
or 16.76 inches
3