7.1 Measurement Of Angles
Objective To find the measure of an angle in
either degrees or radians and to find coterminal
Measurement Of Angles
The angle concept we learned from Geometry must be restated
and generalized since there is no negative angle. In Geometry,
the angle is defined as the inner part formed by two rays of the
In trigonometry, we will learn both positive and negative
angles. In addition, all the angles in trigonometry will be
discussed in a x-y coordinate system. So the definition of
angle must generalized.
A common unit for measuring very large angles is the revolution.
A common unit for measuring smaller angles is the degree of
which there are 360 in one revolution.
Since six equilateral triangles can be arranged within
a circle 1 revolution contains 6 x 60 = 360 degrees.
In trigonometry, the definition of angle is generalized to “The
angle is the result of rotating its initial ray to its terminal ray”.
The side you measure from is called the initial side.
The side you measure to is called the terminal side.
Angles measured counterclockwise are given a positive sign
and angles measured clockwise are given a negative sign.
This is a
Negative Angle counterclockwise
This is a rotation.
clockwise Initial Ray
The Notation to Represent an Angle
It is customary to use small letters in the Greek alphabet
to symbolize angle measurement.
alpha beta gamma
theta phi delta
We can use a coordinate system with angles by putting the
initial side along the positive x-axis with the vertex at the
Quadrant I II
negative Initial Side
If the terminal side is along an Quadrant
axis it is called a quadrantal IV
We say the angle lies in whatever quadrant the terminal side lies
We will be using two different units of measure when talking
about angles: Degrees and Radians
Let’s talk about degrees first. You are probably already
somewhat familiar with degrees.
If we start with the initial side and go all of the
= 360° way around in a counterclockwise direction we
have 360 degrees
If we went 1/4 of the way in a = 90°
clockwise direction the angle
would measure -90°
You are probably already
familiar with a right angle
= - 90° that measures 1/4 of the way
around or 90°
What is the measure of this angle?
You could measure in the positive direction
= - 360o + 45o and go around another rotation which would
be another 360°
= - 315° = 45°
= 360° + 45° = 405°
You could measure in the positive
You could measure in the negative
There are many ways to express the given angle. Whichever
way you express it, it is still a Quadrant I angle since the
terminal side is in Quadrant I.
Angles can be measured more precisely by dividing 1 degree into
60 minutes, and by dividing 1 minute into 60 seconds.
If the angle is not exactly to the next degree it can be expressed
as a decimal (most common in math) or in degrees, minutes and
seconds (common in surveying and some navigation).
1 degree = 60 minutes 1 minute = 60 seconds
To convert to decimal form use conversion fractions. These
are fractions where the numerator = denominator but two
different units. Put unit on top you want to convert to and put
unit on bottom you want to get rid of.
Let's convert the 30" 1' = 0.5'
seconds to minutes
1 degree = 60 minutes 1 minute = 60 seconds
= 25o48'30" = 25o48.5' = 25.808°
Now let's use another conversion fraction to get rid of minutes.
48.5' 1 = .808°
For example, and angle of 25 degrees, 20 minutes, and 6 seconds
is written 25º206
To convert from decimal degrees to degrees, minutes, and seconds:
12.3º = 12º + 0.3(60) = 12º18
To convert from degrees, minutes, and seconds to decimal degrees:
20 6 200 1
25º206 = 25 25 25.335
60 3600 600 600
Another way to measure angles is using what is called radians.
Given a circle of radius r with the vertex of an angle as the center
of the circle, if the arc length formed by intercepting the circle
with the sides of the angle is the same length as the radius r, the
angle measures one radian.
arc length is
r r also r
r initial side
This angle measures
radius of circle is r 1 radian
Arc length s of a circle is found with the following formula:
s = r MEASURE MUST BE IN
RADIANS TO USE FORMULA!
arc length radius measure of angle
Find the arc length if we have a circle with a radius of 3 meters
and central angle of 0.52 radian.
arc length to find is in black
3 s = r = 1.56 m
What if we have the measure of the angle in degrees? We can't
use the formula until we convert to radians, but how?
We need a conversion from degrees to radians. We could use
a conversion fraction if we knew how many degrees equaled
how many radians.
If we look at one revolution
Let's start with
the arc length s = r around the circle, the arc
length would be the
formula circumference. Recall that
2r = r circumference of a circle is
cancel the r's
2 = This tells us that the
radian measure all the
way around is 2. All the
2 radians = 360° way around in degrees is
In general, the radian measure of the central angle, AOB, is the
number of radius units in the length of arc AB
Since the arc length of 1 revolution is the
circumference of the circle:
s 2 r
So 2π radians = 360 degrees , π radians = 180 degrees.
1 radian degrees
1 degree radians
2 radians = 360° radians = 180°
Convert 30°to radians using a conversion fraction.
30 2 radians
The fraction can be
reduced by 2. This
° 360 would be a simpler
180° conversion fraction.
Can leave with or use button on
= radians 0.52
your calculator for decimal.
Convert /3 radians to degrees using a conversion fraction.
radians = 60°
a. Convert 196º to radians (to the nearest hundredth).
b. Convert 1.35 radians to decimal degrees (to the nearest tenth)
and to degrees and minutes (to the nearest ten minutes).
a. 196 196 3.42 radians
b. 1.35 radians 1.35 77.3 7720'
Angle measures that can be expressed evenly in degrees cannot be
expressed evenly in radians, and vice versa.
That is why angles measured in radians are frequently given as
fractional multiples of π.
Angles whose measures are multiples of , , and appear often in trig.
4 3 6
Area of a Sector of a Circle
The formula for the area of a sector of a circle can be derived
from cutting the sector into n smaller sectors of the same vertices.
The area of each smaller sector can be viewed as a slim and tall
triangle with height r and base si .
1 The area of a sector then is
Ai si r the sum of area of all those
n smaller triangles. s1 sn
1 1 n s3
A Ai si r r si
i 1 i 1 2 2 i 1 r
The sum of all n bases for those smaller triangle s i
the length of the arc, which is r. Therefore, i 1
1 1 2
A r (r ) r
Area of a Sector of a Circle
The formula for the area of a
sector of a circle is:
Again must be in RADIANS so if it is 2
in degrees you must convert to radians
to use the formula.
Find the area of the sector if the radius is 3 feet and = 50°
radians 1 2
50 = 0.873 radians A 3 0.837
3.77 sq ft
When an angle is shown in a coordinate plane, it usually appears in
standard position, with its vertex at the origin and its initial ray along the
We consider a counterclockwise rotation to be positive and a clockwise
rotation to be negative.
If the terminal ray of an angle in standard position lies in the first
quadrant, the angle is said to be a first-quadrant angle.
If the terminal ray of an angle in standard position lies along an axis, the
angle is called a quadrantal angle.
Two angles in standard position are called coterminal angles if they
have the same terminal ray.
For any given angle there are infinitely many coterminal angles.
Example 2. Find two angles, one positive and one negative, that are
coterminal with the angle Sketch all three angles.
Convert each degree measure to radians. Leave answers in terms of .
a. 180 b. 90 c. 315 d . 60
e. 120 f . 240 g. 30 h. 1
a. 180 b. 90 c. 315
180 180 2 180 4
d . 60 e. 120 f . 240
180 3 180 3 180 3
g. 30 h. 1
180 6 180 180
Convert each radian measure to degrees.
a. 2 b. c. d.
3 5 11 5
e. f. g. h.
4 3 6 6
360 b. 180 c. 90
180 3 180 5 180
d. 45 e. 135 f. 300
4 4 3
11 180 5 180
g. 330 h. 150
Find two angles, one positive and one negative, that are coterminal with
each given angle.
a. 10 b. 100 c. 5 d . 400
e. f. g. h. 4
a. 370 and 350 b. 460 and 260
c. 355 and 365 d . 40 and 320
e. 3 and f. and
g. and h. 2
A Sense of Angle Sizes
See if you can guess
the size of these
45 angles first in degrees 60
4 and then in radians. 3
90 150 3
2 6 135
You will be working so much with these angles, you
should know them in both degrees and radians.
P. 261 #1 – 8, 9 – 31 (odd),
10 – 18 (even), 22 – 32 (even)