v =
t
Example 11
A water wheel completes 1 rotation every 5 seconds. Find the
angular velocity in radians per second. 3
The wheel completes 1 rotation = 2π radians in 5 seconds, so the
π
angular velocity would be
2
ω =
≈ 1.257 radians per second.
5
Combining the definitions above with the arclength equation, s= rθ, we can find a
relationship between angular and linear velocities. The angular velocity equation can be
solved for θ , giving θ = ω t . Substituting this into the arclength equation gives
s = rθ = rω t .
2 http://commons.wikimedia.org/wiki/File:Pivot_otech_002.JPG CC-BY-SA 3 http://en.wikipedia.org/wiki/File:R%C3%B6mische_S%C3%A4gem%C3%BChle.svg CC-BY
316 Chapter 5
Substituting this into the linear velocity equation gives
s rω t
v = =
= rω
t
t
Relationship Between Linear and Angular Velocity
When the angular velocity is measured in radians per unit time, linear velocity and
angular velocity are related by the equation
v = rω
Example 12
A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are
rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is travelling
down the road.
Here we have an angular velocity and need to find the corresponding linear velocity,
since the linear speed of the outside of the tires is the speed at which the bicycle travels
down the road.
We begin by converting from rotations per minute to radians per minute. It can be
helpful to utilize the units to make this conversion
rotations 2π radians
radians
180
⋅
= 360π
minute
rotation
minute
Using the formula from above along with the radius of the wheels, we can find the
linear velocity
radians
inches
v = (14 inches) 360π
=
5040π
minute
minute
You may be wondering where the “radians” went in this last equation. Remember that
radians are a unitless measure, so it is not necessary to include them.
Finally, we may wish to convert this linear velocity into a more familiar measurement,
like miles per hour.
inches
1 feet
1 mile 60 minutes
5040π
⋅
⋅
⋅
= 14.99 miles per hour (mph).
minute 12 inches 5280 feet
1 hour
Try it Now
7. A satellite is rotating around the earth at 27,359 kilometers per hour at an altitude of
242 km above the earth. If the radius of the earth is 6378 kilometers, find the angular
velocity of the satellite.
Section 5.2 Angles 317
Important Topics of This Section
Degree measure of angle
Radian measure of angle
Conversion between degrees and radians
Common angles in degrees and radians
Coterminal angles
Arclength
Area of a sector
Linear and angular velocity
Try it Now Answers
1. α =
°
150
2. β = °
60
3.
°
126
4. 7π
6 π
5. 215 ≈ 37 525
.
18
6.
°
43
.
107
7. 4.1328 radians per hour
318 Chapter 5
Section 5.2 Exercises
π
π
1. Indicate each angle on a circle: 30°, 300°, -135°, 70°, 2 , 7
3
4
π
π
2. Indicate each angle on a circle: 30°, 315°, -135°, 80°, 7 , 3
6
4
3. Convert the angle 180° to radians.
4. Convert the angle 30° to radians.
π
5. Convert the angle 5 from radians to degrees.
6
π
6. Convert the angle 11 from radians to degrees.
6
7. Find the angle between 0° and 360° that is coterminal with a 685° angle.
8. Find the angle between 0° and 360° that is coterminal with a 451° angle.
9. Find the angle between 0° and 360° that is coterminal with a -1746° angle.
10. Find the angle between 0° and 360° that is coterminal with a -1400° angle.
π
11. Find the angle between 0 and 2π in radians that is coterminal with the angle 26 .
9
π
12. Find the angle between 0 and 2π in radians that is coterminal with the angle 17 .
3
π
13. Find the angle between 0 and 2π in radians that is coterminal with the angle 3
−
.
2
π
14. Find the angle between 0 and 2π in radians that is coterminal with the angle 7
−
.
6
15. On a circle of radius 7 miles, find the length of the arc that subtends a central angle of
5 radians.
16. On a circle of radius 6 feet, find the length of the arc that subtends a central angle of 1
radian.
Section 5.2 Angles 319
17. On a circle of radius 12 cm, find the length of the arc that subtends a central angle of
120 degrees.
18. On a circle of radius 9 miles, find the length of the arc that subtends a central angle of
800 degrees.
19. Find the distance along an arc on the surface of the Earth that subtends a central angle
of 5 minutes (1 minute = 1/60 degree). The radius of the Earth is 3960 miles.
20. Find the distance along an arc on the surface of the Earth that subtends a central angle
of 7 minutes (1 minute = 1/60 degree). The radius of the Earth is 3960 miles.
21. On a circle of radius 6 feet, what angle in degrees would subtend an arc of length 3
feet?
22. On a circle of radius 5 feet, what angle in degrees would subtend an arc of length 2
feet?
23. A sector of a circle has a central angle of 45°. Find the area of the sector if the radius
of the circle is 6 cm.
24. A sector of a circle has a central angle of 30°. Find the area of the sector if the radius
of the circle is 20 cm.
25. A truck with 32-in.-diameter wheels is traveling at 60 mi/h. Find the angular speed of
the wheels in rad/min. How many revolutions per minute do the wheels make?
26. A bicycle with 24-in.-diameter wheels is traveling at 15 mi/h. Find the angular speed
of the wheels in rad/min. How many revolutions per minute do the wheels make?
27. A wheel of radius 8 in. is rotating 15°/sec. What is the linear speed v, the angular
speed in RPM, and the angular speed in rad/sec?
28. A wheel of radius 14 in. is rotating 0.5 rad/sec. What is the linear speed v, the angular
speed in RPM, and the angular speed in deg/sec?
29. A CD has diameter of 120 millimeters. When playing audio, the angular speed varies
to keep the linear speed constant where the disc is being read. When reading along
the outer edge of the disc, the angular speed is about 200 RPM (revolutions per
minute). Find the linear speed.
30. When being burned in a writable CD-R drive, the angular speed of a CD is often
much faster than when playing audio, but the angular speed still varies to keep the
linear speed constant where the disc is being written. When writing along the outer
edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per
minute). Find the linear speed.
320 Chapter 5
31. You are standing on the equator of the Earth (radius 3960 miles). What is your linear
and angular speed?
32. The restaurant in the Space Needle in Seattle rotates at the rate of one revolution per
hour. [UW]
a) Through how many radians does it turn in 100 minutes?
b) How long does it take the restaurant to rotate through 4 radians?
c) How far does a person sitting by the window move in 100 minutes if the radius of
the restaurant is 21 meters?
Section 5.3 Points on Circles Using Sine and Cosine 321
Section 5.3 Points on Circles Using Sine and Cosine
While it is convenient to describe the location of a point on a circle using an angle or a
distance along the circle, relating this information to the x and y coordinates and the circle equation we explored in Section 5.1 is an important application of trigonometry.
A distress signal is sent from a sailboat during a storm, but the transmission is unclear
and the rescue boat sitting at the marina cannot determine the sailboat’s location. Using
high powered radar, they determine the distress signal is coming from a distance of 20
miles at an angle of 225 degrees from the marina. How many miles east/west and
north/south of the rescue boat is the stranded sailboat?
In a general sense, to investigate this, we begin by
drawing a circle centered at the origin with radius r,
( x, y)
and marking the point on the circle indicated by some
angle θ. This point has coordinates ( x, y).
r
If we drop a line segment vertically down from this
θ
point to the x axis, we would form a right triangle
inside of the circle.
No matter which quadrant our angle θ puts us in we
can draw a triangle by dropping a perpendicular line
segment to the x axis, keeping in mind that the values
of x and y may be positive or negative, depending on the quadrant.
Additionally, if the angle θ puts us on an axis, we simply measure the radius as the x or y with the other value being 0, again ensuring we have appropriate signs on the coordinates
based on the quadrant.
Triangles obtained from different radii will all be similar triangles, meaning
corresponding sides scale proportionally. While the lengths of the sides may change, the
ratios of the side lengths will always remain constant for any given angle.
To be able to refer to these ratios more easily, we will give them names. Since the ratios
depend on the angle, we will write them as functions of the angle θ .
Sine and Cosine
For the point ( x, y) on a circle of radius r at an angle of θ , we
can define two important functions as the ratios of the sides of
( x, y)
the corresponding triangle:
r
The sine function:
y
sin(θ ) =
y
r
θ
The cosine function:
x
cos(θ ) =
x
r
322 Chapter 5
In this chapter, we will explore these functions using both circles and right triangles. In
the next chapter we will take a closer look at the behavior and characteristics of the sine
and cosine functions.
Example 1
The point (3, 4) is on the circle of radius 5 at some angle θ. Find cos(θ ) and sin(θ ) .
Knowing the radius of the circle and coordinates of the point, we can evaluate the
cosine and sine functions as the ratio of the sides.
x 3
cos(θ ) = =
y 4
sin(θ ) = =
r 5
r
5
There are a few cosine and sine values which we can determine fairly easily because the
corresponding point on the circle falls on the x or y axis.
Example 2
Find co 90
s(
)
° and sin 90
(
)
°
(0, r)
On any circle, the terminal side of a 90 degree angle
points straight up, so the coordinates of the
r
corresponding point on the circle would be (0, r).
90Ůsing our definitions of cosine and sine,
x 0
co 90
s(
)
° = = = 0
r
r
sin 90
(
)
° = y = r = 1
r
r
Try it Now
1. Find cosine and sine of the angle π .
Notice that the definitions above can also be stated as:
Coordinates of the Point on a Circle at a Given Angle
On a circle of radius r at an angle of θ , we can find the coordinates of the point ( x, y) at that angle using
x = r cos(θ)
y = r sin(θ )
On a unit circle, a circle with radius 1, x =cos(θ) and y =sin(θ).
Section 5.3 Points on Circles Using Sine and Cosine 323
Utilizing the basic equation for a circle centered at the origin, 2
2
2
x + y = r , combined
with the relationships above, we can establish a new identity.
2
2
2
x + y = r
substituting the relations above,
2
2
2
( r cos(θ )) + ( r sin(θ )) = r
simplifying,
2
2
2
2
2
r (cos(θ )) + r (sin(θ )) = r
dividing by 2
r
(cos(θ ))2 + (sin(θ ))2 = 1
or using shorthand notation
cos2 (θ ) + sin2 (θ ) = 1
Here cos2 (θ ) is a commonly used shorthand notation for
2
(cos(θ )) . Be aware that many
calculators and computers do not understand the shorthand notation.
In Section 5.1 we related the Pythagorean Theorem 2
2
2
a + b = c to the basic equation of
a circle 2
2
2
x + y = r , which we have now used to arrive at the Pythagorean Identity.
Pythagorean Identity
The Pythagorean Identity. For any angle θ, cos2 (θ ) + sin2 (θ ) = 1.
One use of this identity is that it helps us to find a cosine value of an angle if we know
the sine value of that angle or vice versa. However, since the equation will yield two
possible values, we will need to utilize additional knowledge of the angle to help us find
the desired value.
Example 3
If
3
sin(θ ) = and θ is in the second quadrant, find cos(θ ) .
7
Substituting the known value for sine into the Pythagorean identity,
2
3 2
cos (θ )
+ = 1
7
9
cos2 (θ ) +
= 1
49
40
cos2 (θ ) =
49
40
40
2 10
cos(θ ) = ±
= ±
= ±
49
7
7
Since the angle is in the second quadrant, we know the x value of the point would be
negative, so the cosine value should also be negative. Using this additional information,
we can conclude that
2 10
cos(θ ) = −
.
7
324 Chapter 5
Values for Sine and Cosine
At this point, you may have noticed that we haven’t found any cosine or sine values from
angles not on an axis. To do this, we will need to utilize our knowledge of triangles.
π
First, consider a point on a circle at an angle of 45 degrees, or .
4
( x, y) = ( x, x)
At this angle, the x and y coordinates of the corresponding point
on the circle will be equal because 45 degrees divides the first
1
y
quadrant in half. Since the x and y values will be the same, the
45°
sine and cosine values will also be equal. Utilizing the
x
Pythagorean Identity,
π
π
cos2
+ sin 2
= 1
since the sine and cosine are equal, we can
4
4
substitute sine with cosine
π
π
cos2
+ cos2
= 1
add like terms
4
4
2cos2 π
= 1
divide
4
π 1
cos2 =
since the x value is positive, we’ll keep the positive root
4 2
π
1
cos =
often this value is written with a rationalized denominator
4
2
Remember, to rationalize the denominator we multiply by a term equivalent to 1 to get
rid of the radical in the denominator.
π
1 2
2
2
cos =
=
=
4
2 2
4
2
Since the sine and cosine are equal,
π
2
sin =
as well.
4
2
The ( x, y) coordinates for a point on a circle of radius 1 at an angle of 45 degrees are
2 2
,
.
2
2
Section 5.3 Points on Circles Using Sine and Cosine 325
Example 4
π
Find the coordinates of the point on a circle of radius 6 at an angle of .
4
Using our new knowledge that
π
2
sin
π
2
=
and cos =
, along with our
4
2
4
2
relationships that stated x = r cos(θ ) and y = r sin(θ ) , we can find the coordinates of the point desired:
π
2
x = 6cos = 6
= 3 2
4
2
π
2
y = 6sin = 6
= 3 2
4
2
Try it Now
2. Find the coordinates of the point on a circle of radius 3 at an angle of °
90 .
Next, we will find the cosine and sine at an angle of
π
30 degrees, or . To do this, we will first draw a
6
( x, y)
triangle inside a circle with one side at an angle of 30
r
degrees, and another at an angle of -30 degrees. If the
30°
resulting two right triangles are combined into one
large triangle, notice that all three angles of this larger
triangle will be 60 degrees.
r
60° y
60°
y
r
60°
Since all the angles are equal, the sides will all be equal as well. The vertical line has
length 2 y, and since the sides are all equal we can conclude that 2 y = r, or
r
y = . Using
2
this, we can find
