Chapter 3
Lesson 3.1
Radian Measure and the Circular Functions
Radian Measure
In most work involving applications of trigonometry, angles are measured in
degrees. In more advanced work in mathematics, the use of radian measure of
angles is preferred. Radian measure also allows us to treat our familiar
trigonometric functions as functions with domains of real numbers, rather than
angles.
The figure shows  in standard position
along with a circle of radius r . The vertex of 
is at the center of the circle. Angle 
intercepts an arc on the circle equal in length to
the radius of the circle. Because of this, angle
 is said to have a measure of one RADIAN.
It follows that an angle of measure 2 radians intercepts an arc equal in length to
1
twice the radius of the circle, an angle of measure radian intercepts an arc
2
equal in length to half the radius of the circle, and so on.
Therefore an angle of measure 360 would correspond to a complete circle and
intercepts an arc equal in length to the distance around the circle called the
circumference or 2 times the radius of the circle. Because of this, an angle of
360 has a measure of 2 radians.
2 radians  360
An angle of 180 is half the size of an angle of 360 , so an angle of 180 has half
the radian measure of an angle of 360 .
 radians  180
We can use this simple relationship to convert from degree measure to radian
measure or from radians to degrees.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
EX:
Convert each degree measure to radians.
a)
45
RULE:
b)
Multiply degrees by
 rads
180
330
Convert each radian measure to degrees.
d)
9
4
RULE:
Multiply radians by
120
to convert to radians.
EX:
e)
c)
5
3
f)
7
6
180
to convert to degrees.
 rads
Keep in mind decimal degrees can also be converted the same way, just round
to the nearest thousandths.
CAUTION Figure 2 shows angles measuring 30radians and 30 . These angle
measures are not at all close, so be careful not to confuse them.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
Common radian measures of the reference and quadrantal angles in degrees are
used in this chapter. Trigonometric function values for angles measured in
radians can be found by first converting the radian measure to degrees. (You
should try to skip this intermediate step ASAP, and find the function values
directly from the radian measure.)
Degrees
0
Radians
0
30

6

4

3

2

45
60
90
180
3
2
2
270
360
EX:
Find the TRIG function value.
g)
tan
2
3
h)
sin
3
2
i)
csc
j)
cos
3
4
k)
cot
 5
6
l)
sec
Trigonometry Chapter 3
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
3
11
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Radian Measure and the Circular Functions
Lesson 3.2
Applications of Radian Measure
Radian measure is used to simplify certain formulas. Two of these formulas are
discussed in this section. Both would be more complicated if expressed in
degrees.
ARC LENGTH OF A CIRCLE
To find the length of an arc on a circle depends on the fact that the length of an
arc is proportional to the measure of its central angle.
QOP has a measure of 1 radian and cuts an
arc of length r on the circle.
ROT has a measure of  radians and cuts
an arc of length s on the circle.
Since the lengths of the arcs are proportional
to the measure of their central angles.
s 

 s  r
r 1
The length of the arc s cut on a circle of radius r by a central angle of measure
 is given by the product of the radius and the radian measure of the angle.
CAUTION: When applying the formula s  r , the value of  MUST be
expressed in RADIANS.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
EX: A circle has a radius of 18.2cm . Find the length of the arc cut by a central
angle having the following measures:
a)

3
8
b)
  144
EX: Find the length of the arc intercepted by a central angle  in a circle of
radius r .
c)
r  12.3cm
2

3
d)
r  4.82m
  60
SECTOR OF A CIRCLE
To find the area of a “piece of pie” or sector of a circle is the portion of the interior
of a circle cut by a central angle.
To find the area of a sector, assume that the
radius of the circle is r . A complete circle has
a measure of 2 radians. If a central angle for
the sector has a measure  radians, then the

sector makes up a fraction
of a complete
2
circle. The area of a complete circle is A  r 2 .
Therefore the area of the sector is given by the product of the fraction

and
2
the total area.
A
Trigonometry Chapter 3

1
 r 2  A   r 2
2
2
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Radian Measure and the Circular Functions
CAUTION: As in the formula for arc length, the measure of  must be in
radians when using the formula for the area of a sector.
EX:
Find the area of each of the following sectors of a circle.
r  9.0m
e)


f)
3
r  52cm
3

10
g)
r  12.7cm
  81
EX: Find the distance in kilometers between each of the following pairs of
cities whose latitudes are given. Assume that the cities are on a north-south line
and that the radius of the earth is 6400km.
h)
Madison, South Dakota
Dallas, Texas
44N
33N
EX: The field is in the shape of a sector of a circle. The central angle is 15
and the radius of the circle is 321m. Find the area of the field.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
Lesson 3.3
Circular functions of Real Numbers
So far we have defined the six trigonometric functions for angles. The angles
can be measured either in degrees or in radians. While the domain of the
trigonometric functions is a set of angles, the range is a set of real numbers. In
advanced work, it is necessary to modify the trigonometric functions so that the
domain contains not angles, but real numbers. To do this we use the relationship
between an angle  and an arc of length s on a circle.
In the figure, starting at the point 1,0 , we lay off an arc of length s along the
circle. We go counterclockwise if s is positive and clockwise if s is negative. The
endpoint of the arc is the point x, y  . The circle with center at the origin and a
radius of one unit is called a UNIT CIRCLE. Recall the equation of this circle is
x2  y 2  1.
Radian measure of  is related
to the arc length of s , by the
equation s  r . Here, r  1 , so s ,
which is measured in linear units
such as inches or centimeters, is
numerically equal to  , measured
in radians. Thus, the
trigonometric functions of angle
 in radians found by choosing a
point x, y  on the unit circle can
be rewritten as functions of the
arc length s , a real number. To
distinguish these from the
trigonometric functions of angles,
they are called CIRCULAR
FUNCTIONS.
CIRCULAR FUNCTIONS
sin s  y
cos s  x
Trigonometry Chapter 3
y
x
x
cot s 
y
1
x
1
csc s 
y
tan s 
sec s 
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Radian Measure and the Circular Functions
EX:
Find the EXACT circular function value for each of the following.
a)
cos
2
3
b)
  
tan 

 4 
c)
csc
10
3
You can use a calculator to find an approximation of a circular function of a real
number. But first you must set the calculator to RADIAN MODE. You will use
the same technique from Chapter 2.
EX:
Find an approximation for each circular function value.
d)
cos 0.5149
e)
cot 1.3209
f)
sec(2.9234)
Now let’s find the value of s (arc length not angle) that will give us the circular
function value given. You will use the same technique as before to find the angle
except we are given an interval to stay between.
EX:
 
Find the value of s in the interval 0,  that has the circular function
 2
value.
g)
cos s  0.96854556
h)
cot s  0.29949853
EX: Find the EXACT value of s in the given interval that has the given circular
function value. Do NOT use a calculator.
i)
sin s 
2  
, ,
2  2 
Trigonometry Chapter 3
j)
8
 3 
tan s  3,  , 
 2 
Radian Measure and the Circular Functions
Lesson 3.4
Linear and Angular Velocity
In many situations it is necessary to know at what speed a point on a circular disk
is moving or how fast the central angle of such a disk is changing. Some
examples occur with machinery involving gears or pulleys or the speed of a car
around a curved portion of a highway.
Suppose that point P moves at a constant
speed along a circle of radius r and center O .
The measure of how fast the position of P is
changing is called LINEAR VELOCITY.
Velocity 
distance
s
v
time
t
where s  length of the arc traced by point P at
time t (this formula is just a restatement of the
familiar result d  rt with s as distance, v as the
rate and r as time)
As point P moves along the circle, ray OP rotates around the origin. Since the
ray OP is the terminal side of POB , the measure of the angle changes as P
moves along the circle. The measure of how fast POB is changing is called
ANGULAR VELOCITY.


t
where  is the measure of POB at time t
As with the earlier formulas in this chapter,  must be measured in radians, with
 expressed as radians per unit of time. Angular velocity is used in physics and
engineering, among other applications.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
Linear Velocity also can be written in 2 more ways.
v
s
r

, but since s  r we get v 
and since   we get v  r
t
t
t
The last formula relates linear and angular velocity.
EX:
Suppose that point P is on a circle with radius of 10 centimeters and ray

radians per second.
OP is rotating with angular velocity of
18
a)
Find the angle generated by P in 6 seconds.
b)
Find the distance traveled by P along the circle in 6 seconds.
c)
Find the linear velocity of P .
In practical applications, angular velocity is often given as revolutions per unit of
time, which must be converted to radians per unit of time before using the
formulas given in this section. 1 Revolution = 2 will be used in the conversion.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions
EX:
A belt runs a pulley of radius 6 centimeters at 80 revolutions per minute.
a)
Find the angular velocity of the pulley in radians per second.
b)
Find the linear velocity of the belt in centimeters per second.
EX: A satellite traveling in a circular orbit 1600 kilometers above the surface of
the Earth takes two hours to make an orbit. Assume that the radius of the Earth
is 6400 kilometers.
a)
Find the linear velocity of the satellite.
b)
Find the distance traveled in 4.5 hours.
Trigonometry Chapter 3
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Radian Measure and the Circular Functions