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Section 7-1
Measurement of Angles
Trigonometry

The word trigonometry comes two Greek
words, trigon and metron, meaning
“triangle measurement.”
Trigonometry

In trigonometry, an angle often
represents a rotation about a point.
Thus, the angle Θ shown is the result of
rotating its initial ray to its terminal ray.
Revolutions and Degrees


A common unit for measuring very large
angles is the revolution, a complete
circular motion.
A common unit for measuring smaller
angles is the degree, of which there are
360 in one revolution
Degrees, Minutes, and Seconds

Angles can be measured more precisely
by dividing one degree into 60 minutes
and by dividing one minute into 60
seconds. For example, an angle of 25
degrees, 20 minutes, and 6 seconds is
written 25˚20’6”.
Decimal Degrees



Angles can also be measured in decimal
degrees. To convert between decimal
degrees and degrees, minutes, and
seconds, you can reason as follows:
12.3˚=12˚+ 0.3(60)’ = 12˚18’
 20   6 
25˚20’6”=25˚ +  60    3600   25.335
Radians

Relatively recently in mathematical history,
another unit of angle measurement, the radian,
has come into use. When an arc of a circle has
the same length as the radius of the circle, the
measure of the central angle AOB , is by
definition 1 radian.
Radians

Likewise, a central angle has a measure of
1.5 radians when the length of the
intercepted arc is 1.5 times the radius.
Radian Measures

In general, the radian measure of the
central angle AOB is the number of
radius units in the length of arc AB. This
accounts for the name radian. In the
diagram at the right, the measure (Greek
s
Theta) of the central angle is:  
r
Arc Length

Arc length = s = r 
Radians

Let us use this equation to see how
many radians correspond to 1 revolution.
Since the arc length of 1 revolution is the
circumference of the circle, 2Πr, we have
s 2r
 
 2 . Thus, 1 revolution
r
r
measured in radians is 2Π and
measured in degrees is 360. We have
2Π radians = 360 degrees or Π radians =
180 degrees.
Conversion Formulas


This gives us the following conversion
formulas:
180
1 radian =
degrees ≈ 57.2958
degrees 
Conversion Formulas


This gives us the following conversion
formulas:

1 degree =
radians ≈0.0174533
180
radians
Radians

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
often given as fractional multiples of Π.
Degrees vs. Radians

Angles whose measures are multiples of
 

, , and appear often in trigonometry. The
4 3
6
diagrams below will help you keep the degree
conversions for these special angles in mind.
Note that a degree measure, such as 45˚, is
usually written with a degree symbol, while a
radian measure such as  is usually written
4
without any symbol.
45°

 Multiples of 45°,
4
60°


Multiples of 60°, 3
30°

 Multiples of 30°,
6
Standard Position

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 positive x-axis.
We consider a counterclockwise rotation
to be positive and a clockwise rotation to
be negative. By positive and negative
angles we mean angles with positive and
negative measures.
Positive Angle

An angle of 380˚
Negative Angle


An angle of 
2
Quadrantal Angles

If the terminal ray of an angle in standard
position lies in the first quadrant, as shown at
the left above, the angle is said to be a firstquadrant angle. Second-, third-, and fourthquadrant angles are similarly defined. If the
terminal ray of an angle in standard position
lies along an axis, as shown at the right above,
the angle is called a quadrantal angle. The
measure of a quadrantal angle is always a
multiple of 90˚ or 
2
Coterminal Angles

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


Convert the degree measures to radians.
270°
-23.6°
Example

Convert the radian measures to degrees
12.3
Example





Find two angles, one positive and one
negative, that are coterminal with each
given angle.
A. 60°
B. -210°
C.
D.
Example

Find two angles, one positive and one
negative, that are coterminal with each
given angle.
24°15’
-23°37’
Additional Example:
1.
Give the degree measure and the
radian measure of the angle formed by
the hour hand and the minute hand of a
clock at 2:30 a.m.
Additional Example:
2. A gear revolves at 40 rpm. Find the
number of degrees per minute through
which the gear turns and the
approximate number of radians per
minute.