Download M19500 PreCalculus Chapter 6.1: Angle Measure

Angle Measure
M19500 PreCalculus Chapter 6.1: Angle Measure
Tamara Kucherenko
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angle Measure
B
terminal
side
O
A
initial side
An angle AOB consists of two rays OA and OB with a common
vertex O.
We interpret an angle as a rotation of the ray OA onto OB.
OA is the initial side and OB is the terminal side of the angle.
The angle is positive if the rotation is counter-clockwise.
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angle Measure
B
terminal
side
O
A
initial side
O
A
An angle AOB consists of two rays OA and OB with a common
vertex O.
We interpret an angle as a rotation of the ray OA onto OB.
OA is the initial side and OB is the terminal side of the angle.
The angle is positive if the rotation is counter-clockwise.
The angle is negative, if the rotation is clockwise.
B
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angle Measure
B
terminal
side
O
A
initial side
O
A
B
An angle AOB consists of two rays OA and OB with a common
vertex O.
We interpret an angle as a rotation of the ray OA onto OB.
OA is the initial side and OB is the terminal side of the angle.
The angle is positive if the rotation is counter-clockwise.
The angle is negative, if the rotation is clockwise.
The measure of an angle is the amount of rotation about the
vertex required to move OA onto OB.
One unit of measurement for angles is the degree.
An angle of measure 1 degree is formed by rotating the initial
1
side 360
of a complete revolution.
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angle Measure
Another unit of measurement for angles is the radian.
Definition of Radian Measure
If a circle of radius 1 is drawn with the vertex of an angle at its center, then the
measure of this angle in radians (abbreviated rad) is the length of the arc that
subtends the angle.
The circumference of the circle of radius 1 is 2π. Thus, a complete revolution has
measure 2π rad, a straight angle has measure π rad, a right angle has measure π2 rad.
π
2
π rad
O
O
Tamara Kucherenko
rad
1 rad
O
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angle Measure
Relationship between degrees and radians
180 ◦
◦
180 = π rad 1 rad =
π
1
To convert degrees to radians, multiply by
π
.
180
2
To convert radians to degrees, multiply by
180
.
π
Example 1. Express 30◦ in radians.
π π
Solution. 30◦ = 30
rad = .
180
6
Tamara Kucherenko
1◦ =
π
rad
180
Example 2. Express π/3 rad in degrees.
π 180 π
Solution. rad =
= 60◦ .
3
3
π
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
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.
y
y
x
O
y
x
O
x
O
Two angles in standard position are coterminal if their sides coincide.
The last two angles above are coterminal.
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Angles in Standard Position
Thus angles are coterminal if their difference is a whole number of circles: either a
whole number times ±2π rad or a whole number times ±360 degrees.
7π
Example 1. Are angles − 5π
3 and 3 coterminal?
Solution. We look at the difference of the two angles: − 5π
3 −
which is 2 · −2π. Yes, the angles are coterminal.
7π
3
= − 12π
3 = −4π,
To find angles that are coterminal with a given angle , add or subtract a whole number
times 360 degrees, or a whole number times 2π rad.
Example 2. Find an angle with measure between 0◦ and 360◦ that is coterminal with
the angle of measure 1290◦ in standard position.
Solution. We can subtract 360◦ as many times as we wish from 1290◦ , and the
resulting angle will be coterminal with 1290◦ . Thus, 1290◦ − 360◦ = 930◦ is
coterminal with 1290◦ , but it is not between 0◦ and 360◦ . Just keep subtracting 360◦ :
1290◦ − 360◦ − 360◦ − 360◦ = 210◦ .
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Length of a Circular Arc
s
θ
O
r
An angle whose radian measure is θ is subtended by an arc that
is the fraction of the circumference of a circle. Thus, in a circle
of radius r, the length s of an arc that subtends the angle θ is
s=
θ
2π (circumference
of circle) =
θ
2π (2πr)
= θr
Length of a circular arc
In a circle of radius r, the length s of an arc that subtends a central angle of θ rad is
s = rθ
Example 3: Find the length of an arc of a circle with radius 10 meters that subtends a
central angle of 30◦ .
π
=
Solution. First, we convert 30◦ into radians: 30 180
The length of the arc is s = rθ =
(10) π6 =
Tamara Kucherenko
5π
3
π
6
rad.
meters .
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Area of a Circular Sector
s
θ
O
r
The area of a circle of radius r is a = πr2 . A sector of this circle
with central angle θ has an area that is the fraction of the area
of the entire circle. So the area of this sector is
A=
θ
2π (area
of circle) =
θ
2
2π (πr )
= 12 r2 θ
Area of a circular sector
In a radius r circle, the area A of a sector with central angle θ rad is A = 12 r2 θ
Example 4: Find the area of a sector of a circle with central angle 60◦ if the radius of
the circle is 3 meters.
π
Solution. First, we convert 60◦ into radians: 60 180
=
1
π
3π
1 2
2
A = 2 r θ = 2 (3) 3 = 2 square meters.
Tamara Kucherenko
π
3
rad. So the area of the arc is
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Circular Motion: Optional for Math 19500 Fall 2013
s
θ
O
r
There are two ways to describe the motion of a point moving at
constant speed along a circle.
distance traveled along the circle
Linear speed =
.
time
Angular speed =
change of angle (in radians)
.
time
Linear speed and angular speed
Suppose a point moves along a circle of radius r and the ray from the center of the
circle to the point traverses θ radians in time t. Let s = rθ be the distance the point
travels in time t. Then the speed of the object is given by
Angular speed ω = θt
Linear speed v = st
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Circular Motion: Optional for Math 19500 Fall 2013
Example 5: 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 seconds, the angle θ changes by 15 · 2π = 30π radians.
θ
t
So the angular speed of the stone is ω =
=
30π rad
10 seconds
= 3π rad/second.
Distance traveled by the stone in 10 seconds is s = 15 · 2πr = 15 · 2π · 3 = 90π feet.
So the linear speed of the stone is v =
s
t
=
Tamara Kucherenko
90π feet
10 seconds
= 9π feet/second .
M19500 PreCalculus Chapter 6.1: Angle Measure
Angle Measure
Circular Motion: Optional for Math 19500 Fall 2013
Relationship between linear and angular speed
If a point moves along a circle of radius r with angular speed v, then its linear speed v
is given by v = rω .
Example 6: A woman is riding a bicycle whose wheels are 22 inches in diameter. If
the wheels rotate at 125 revolutions per minute find the speed at which she is
traveling, in miles per hour.
Solution. Each revolution is 2π radians, and so the wheels’ angular speed is
2π · 125 = 250 rad/min. Since the wheels have radius 11 inches, the linear speed is
v = rω = 11 · 250π inches/minute.
To convert inches/minute to miles/hour recall that
1 mile = 5280 feet = 5280 · 12 inches, while 1 hour = 60 minutes. Thus
1 mile
60 minutes
inches
= 11 · 250π inches
= 11·250π·60
11 · 250π minute
minute · 5280·12 inches ·
hour
88·60·12 =
Tamara Kucherenko
M19500 PreCalculus Chapter 6.1: Angle Measure
125π miles
48 hour