Trigonometric Functions:
6 Right Triangle Approach
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6.1 Angle Measure
Copyright © Cengage Learning. All rights reserved.
Objectives
■ Angle Measure
■ Angles in Standard Position
■ Length of a Circular Arc
■ Area of a Circular Sector
■ Circular Motion
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Angle Measure
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Angle Measure
An angle AOB consists of two rays R1 and R2 with a
common vertex O (see Figure 1).
We often interpret an angle as a rotation of the ray
R1 onto R2.
Positive angle Negative angle
Figure 1
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Angle Measure
In this case R1 is called the initial side, and R2 is called the
terminal side of the angle.
If the rotation is counterclockwise, the angle is considered
positive, and if the rotation is clockwise, the angle is
considered negative.
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Angle Measure
The measure of an angle is the amount of rotation about
the vertex required to move R1 onto R2.
Intuitively, this is how much the angle “opens.” One unit of
measurement for angles is the degree.
An angle of measure 1 degree is formed by rotating the
initial side of a complete revolution.
In calculus and other branches of mathematics a more
natural method of measuring angles is used: radian
measure.
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Angle Measure
Figure 2
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Angle Measure
The circumference of the circle of radius 1 is 2, so a
complete revolution has measure 2 rad, a straight angle
has measure rad, and a right angle has measure /2 rad.
An angle that is subtended by an arc of length 2 along the
unit circle has radian measure 2 (see Figure 3).
Radian measure
Figure 3
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Angle Measure
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Example 1 – Converting Between Radians and Degrees
(a) Express 60 in radians. (b) Express rad in degrees.
Solution:
The relationship between degrees and radians gives
(a) 60
(b)
= 30
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Angle Measure
A note on terminology: We often use a phrase such as
“a 30 angle” to mean an angle whose measure is 30.
Also, for an angle we write = 30 or = /6 to mean the
measure of is 30 or /6 rad.
When no unit is given, the angle is assumed to be
measured in radians.
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Angles in Standard Position
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Angles in Standard Position
An angle is in standard position if it is drawn in the
xy-plane with its vertex at the origin and its initial side on
the positive x-axis.
Figure 5 gives examples of angles in standard position.
(a) (b) (c) (d)
Angles in standard position
Figure 5
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Angles in Standard Position
Two angles in standard position are coterminal if their
sides coincide.
In Figure 5 the angles in (a) and (c) are coterminal.
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Example 2 – Coterminal Angles
(a) Find angles that are coterminal with the angle = 30 in
standard position.
(b) Find angles that are coterminal with the angle = in
standard position.
Solution:
(a) To find positive angles that are coterminal with , we
add any multiple of 360.
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Example 2 – Solution cont’d
Thus
30° + 360° = 390° and 30° + 720° = 750°
are coterminal with = 30. To find negative angles that
are coterminal with , we subtract any multiple of 360°.
Thus
30° – 360° = –330° and 30° – 720° = –690°
are coterminal with .
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Example 2 – Solution cont’d
See Figure 6.
Figure 6
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Example 2 – Solution cont’d
(b) To find positive angles that are coterminal with , we
add any multiple of 2.
Thus
and
are coterminal with = /3.
To find negative angles that are coterminal with , we
subtract any multiple of 2.
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Example 2 – Solution cont’d
Thus
and
are coterminal with . (See Figure 7.)
Figure 7
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Length of a Circular Arc
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Length of a Circular Arc
Solving for , we get the important formula
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Length of a Circular Arc
This formula allows us to define radian measure using a
circle of any radius r : The radian measure of an angle is
s/r, where s is the length of the circular arc that subtends
in a circle of radius r (see Figure 10).
The radian measure of is the number of “radiuses” that
can fit in the arc that subtends ; hence the term
Figure 10
radian.
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Example 4 – Arc Length and Angle Measure
(a) Find the length of an arc of a circle with radius 10 m that
subtends a central angle of 30.
(b) A central angle in a circle of radius 4 m is subtended
by an arc of length 6 m. Find the measure of in
radians.
Solution:
(a) From Example 1(b) we see that 30 = /6 rad. So the
length of the arc is
s = r = = 24
Example 4 – Solution cont’d
(b) By the formula = s/r we have
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Area of a Circular Sector
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Area of a Circular Sector
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Example 5 – Area of a Sector
Find the area of a sector of a circle with central angle 60 if
the radius of the circle is 3 m.
Solution:
To use the formula for the area of a circular sector, we
must find the central angle of the sector in radians:
60° = 60( /180) rad = /3 rad.
Thus the area of the sector is
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Circular Motion
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Circular Motion
Suppose a point moves along a circle as shown in Figure
12. There are two ways to describe the motion of the point:
linear speed and angular speed.
Linear speed is the rate at which the distance traveled is
changing, so linear speed is the distance traveled divided
by the time elapsed.
Figure 12
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Circular Motion
Angular speed is the rate at which the central angle is
changing, so angular speed is the number of radians this
angle changes divided by the time elapsed.
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Example 6 – Finding Linear and Angular Speed
A boy rotates a stone in a 3-ft-long sling at the rate of 15
revolutions every 10 seconds. Find the angular and linear
velocities of the stone.
Solution:
In 10 s the angle changes by 15 2 = 30 rad. So the
angular speed of the stone is
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Example 6 – Solution cont’d
The distance traveled by the stone in 10 s is
s = 15 2 r = 15 2 3 = 90 ft.
So the linear speed of the stone is
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Circular Motion
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Example 7 – Finding Linear Speed from Angular Speed
A woman is riding a bicycle whose wheels are 26 in. in
diameter. If the wheels rotate at 125 revolutions per minute
(rpm), find the speed (in mi/h) at which she is traveling.
Solution:
The angular speed of the wheels is
2 125 = 250 rad/min.
Since the wheels have radius 13 in. (half the diameter), the
linear speed is
= r = 13 250 10,210.2 in./min 35
Example 7 – Solution cont’d
Since there are 12 inches per foot, 5280 feet per mile, and
60 minutes per hour, her speed in miles per hour is
9.7 mi/h
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