Transcript Radian and Degree Measure - William H. Peacock, LCDR USN

4.1
Radian and Degree Measure
Objectives:
1.Describe angles
2.Use radian measure
3.Use degree measure
4.Use angles to model and solve real-life problems
2
Angles
3
Angles
As derived from the Greek language, the word
trigonometry means “measurement of triangles.”
Initially, trigonometry dealt with relationships among the
sides and angles of triangles.
An angle is determined by rotating a ray (half-line) about
its endpoint.
4
Angles
The starting position of the ray is the initial side of the
angle, and the position after rotation is the terminal side,
as shown.
Angle
5
Angles
The endpoint of the ray is the vertex of the angle.
Angle
This perception of an angle fits a
coordinate system in which the origin
is the vertex and the initial side
coincides with the positive x-axis.
Such an angle is in standard position,
as shown.
Angle in standard position
6
Angles
Counterclockwise rotation generates
positive angles and clockwise rotation
generates negative angles.
Angles are labeled with Greek letters
such as  (alpha),  (beta), and  (theta),
as well as uppercase letters A, B, and C.
Note that angles
 and  have the same initial and
terminal sides. Such angles are
coterminal.
Coterminal angles
7
Angle describes the amount and direction of rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
8
Angles - Review
Angle = determined by rotating a ray (half-line) about its
endpoint.
Initial Side = the starting point of the ray
Terminal Side = the position after rotation
Vertex = the endpoint of the ray
Positive Angles = generated by counterclockwise rotation
Negative Angles = generated by clockwise rotation
Angle
Angle in standard position
9
Radian Measure
10
Radian Measure
Measure of an Angle = determined by the amount of rotation
from the initial side to the terminal side (one way to measure
angle is in radians)
Central Angles = an angle whose vertex is the center of the
circle
Central Angle
11
Radian Measure
You determine the measure of an angle by the amount of
rotation from the initial side to the terminal side. One way to
measure angles is in radians. This type of measure is
especially useful in calculus.
To define a radian, you can use
a central angle of a circle, one
whose vertex is the center of the
circle, as shown.
Arc length = radius when  = 1 radian
12
Radian Measure
Given a circle of radius (r) with the vertex of an angle as the
center of the circle, if the arc length (s) formed by intercepting
the circle with the sides of the angle is the same length as the
radius (r), the angle measures one radian.
s
r
r
radius of circle is r
arc length is
also r (s=r)
initial side
This angle measures 1
radian
13
Radian Measure
• The radian measure of a central angle θ is obtained
by dividing the arc length s by r (θ = s/r).
• Because the circumference of a circle is 2 r units,
it follows that a central angle of one full revolution
(counterclockwise) corresponds to an arc length of
s = 2 r.
• Therefore, a full circle measures
2𝜋𝑟
𝑟
, or 2𝜋 radians.
14
Radian Measure
Moreover, because 2  6.28, there are just over six radius
lengths in a full circle, as shown.
Because the units of measure for s and r are the same, the
ratio s/r has no units—it is a real number.
15
Radian Measure
Because the measure of an angle of one full revolution is
s/r = 2 r/r = 2 radians, you can obtain the following.
180 ̊
90 ̊
60 ̊
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Radian Measure
These and other common angles are shown below.
30 ̊
45 ̊
60 ̊
90 ̊
180 ̊
360 ̊
17
Radian Measure
We know that the four quadrants in a coordinate system
are numbered I, II, III, and IV.
The figure to the right shows which
angles between 0 and 2 lie in
each of the four quadrants.
Note that angles between
0 and  /2 are acute angles and
angles between  /2 and  are
obtuse angles.
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Comments on Radian Measure
• A radian is an amount of rotation that is independent
of the radius chosen for rotation
• For example, all of these give a rotation of 1 radian:
1. radius of 2 rotated along an arc length of 2
2. radius of 1 rotated along an arc length of 1
3. radius of 5 rotated along an arc length of 5, etc.
r1
r1
r2 1 rad
r2
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Coterminal Angles
• Two angles are coterminal if they have the same initial and
terminal sides.
• The angles 0 and 2𝜋 are coterminal.
• You can find an angle that is coterminal to a given angle θ
by adding or subtracting 2𝜋.
• For negative angles subtract 2𝜋 and for positive angles add
2𝜋.
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Coterminal Angles
Two angles are coterminal when they have the same initial
and terminal sides. For instance, the angles 0 and 2 are
coterminal, as are the angles  /6 and 13 /6.
You can find an angle that is coterminal to a given angle 
by adding or subtracting 2 (one revolution), as
demonstrated in Example 1.
A given angle  has infinitely many coterminal angles.
For instance,  =  /6 is coterminal with
where n is an integer.
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Example 1 – Finding Coterminal Angles
a. For the positive angle 13 /6, subtract 2 to obtain a
coterminal angle
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Example 1 – Finding Coterminal Angles cont’d
b. For the negative angle –2 /3, add 2 to obtain a
coterminal angle
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Example 2 – Finding Coterminal Angles
Find a positive coterminal angle to 20º
20  360  380
Find a negative coterminal angle to 20º 20  360  340 
15
Find 2 coterminal angles to
(one positive and one negative).
4
15 8
15

 2 
4
4
4
23

4
15
15 8
 2 

4
4
4

7 8



4
4
4
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Your Turn:

Find two Coterminal Angles (+ and -)
13
6
3
4
2

3
positive
negative
25𝜋
6
11𝜋
4
11𝜋
−
6
5𝜋
−
4
4𝜋
3
8𝜋
−
3
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Complementary & Supplementary Angles
Two positive angles  and  are complementary
(complements of each other) when their sum is  /2.
Two positive angles are supplementary (supplements of
each other) when their sum is .
Complementary angles
Supplementary angles
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Example:
Complementary Angles: Two angles whose sum is 90 or (𝜋/2)
radians.

6
3  2 
 
 

2 6
6 6
6
3


Supplementary Angles: Two angles whose sum is 180 or 𝜋
radians.
2
3
2


3
3 2 


3
3
3
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Your Turn:
Two positive angles are complementary if their sum is
𝜋/2. Two positive angles are supplementary if their
sum is 𝜋.
Find the Complement and Supplement
2
5
4
5
𝜋
Complement
10
3𝜋
Supplement
5
Complement - None
𝜋
Supplement
5
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Degree Measure
A second way to measure angles is in degrees, denoted
by the symbol .
A measure of one degree (1) is equivalent to a rotation of
of a complete revolution about the vertex.
To measure angles, it is
convenient to mark degrees on
the circumference of a circle.
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Degree Measure
So, a full revolution (counterclockwise) corresponds to
360, a half revolution to 180, a quarter revolution to 90,
and so on.
Because 2 radians corresponds to one complete
revolution, degrees and radians are related by the
equations
360 = 2 rad and 180 =  rad.
From the latter equation, you obtain
and
which lead to the conversion rules in the next slide.
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Degree Measure
31
Degree Measure
When no units of angle measure are specified, radian
measure is implied.
For instance,  = 2, implies that  = 2 radians.
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Example – Converting from Degrees to Radians
a.
Multiply by  rad / 180.
b.
Multiply by  rad / 180.
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Example – Converting from Radians to Degrees
 8  8 180
a.


3
3

 480
5 5 180
 150
b.


6
6 
Multiply by 180/ rad.
Multiply by 180/ rad.
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Your Turn:
Convert from degrees to radians
135°
3𝜋
4
540°
3𝜋
-270°
3𝜋
2
Convert from radians to degrees


rad
−90°
9
rad
2
2rad
810°
2
360°
≈ 114.59°
𝜋
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Equivalent Angles in Degrees and Radians
Degrees
Radians
Degrees
Exact
Approximate
0
0
0
90
30

6
.52
180
45

4

3
.79
1.05
60
Radians
Exact
Approximate

2

1.57
270
3
2
4.71
360
2
6.28
3.14
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Equivalent Angles in Degrees and Radians cont.
37
A Sense of Angle Sizes
45  
30  
90  

2

4

6
See if you can guess the size
of these angles first in degrees
and then in radians.
2
120  
3
5
150 
6
60  

3
180   
3
135 
4
You will be working so much with these angles, you should know them in
both degrees and radians.
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Applications
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Vocabulary
►Arc
►Sector
►Tangent
►Secant
►Chord
►Segment
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Arc, Sector, Segment
►Arc: any unbroken part of the
circumference
Measured in radians
►Sector: a plane figure bounded by
two radii and the included arc
►Segment: a part cut off from a circle
by a line, as a part of a circular area
contained by an arc and its chord
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Intercepted Arcs
►An intercepted arc is the arc that is formed when
segments intersect portions of a circle and create arcs.
These segments in effect 'intercept' parts of the circle.
42
Major Arcs/Minor Arcs
►The bigger one is major (≥π). The smaller is minor(<π).
►Arc HN is the minor arc
►Arc HKN is the major arc
43
Arc Lengths and Central Angles of a Circle
• Given a circle of radius “r”, any angle with vertex at the
center of the circle is called a “central angle”.
• The portion of the circle intercepted by the central angle
is called an “arc” and has a specific length called “arc
length” represented by “s”.
• From geometry it is know that in a specific circle the
length of an arc is proportional to the measure of its
central angle.
• For any two central angles,  1 and 2 , with
corresponding arc lengths s1 and s 2 : s
s
1
1

2
2
44
Development of Formula for Arc Length
Since this relationship is true for any two central angles
and corresponding arc lengths in a circle of radius r:
s1
1

s2
2
Let one angle be  rad with corresponding arc length
and let the other central angle be 2 rad , a whole
rotation, with arc length 2r .
s
2r

 rad 2 rad
s

r
s
s  r
 in radians!
45
Applications
The radian measure formula,  = s/r, can be used to
measure arc length along a circle.
46
Example – Finding Arc Length
A circle has a radius of 4 inches. Find the length of the arc
intercepted by a central angle of 240.
47
Example – Solution
To use the formula s = r, first convert 240 to radian
measure.
48
Example – Solution
cont’d
Then, using a radius of r = 4 inches, you can find the arc
length to be
s = r
Note that the units for r determine the units for r because 
is given in radian measure, which has no units.
49
Your Turn: Finding Arc Length
A circle has radius 18.2 cm. Find the length of the arc
intercepted by a central angle having the following
measure:
3

8
s  r
 3 
s  18.2 
 cm
 8 
54.6
s
cm  21.4cm
8
50
Your Turn: Finding Arc Length
cont’d
For the same circle with r = 18.2 cm and  = 144, find the
arc length
convert 144 to radians
  
144  144 

 180 
4

radians
5
s  r
 4 
s  18.2 
 cm
 5 
72.8
s
cm  45.7cm
5
51
Your Turn:
s = rθ
(s = arc length, r = radius, θ= central angle
measure in radians)
a. A circle has a radius of 4 inches. Find the length
of the arc intercepted by a central angle of 240°.
16/3𝜋 ≈ 16.8 in
b. A circle has a radius of 8 cm. Find the length of
the arc intercepted by a central angle of 45°
2𝜋 ≈ 6.3 cm
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Applications
A sector of a circle is the region bounded by two radii of
the circle and their intercepted arc.
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Applications
54
Example:
Find the area of the sector of a circle of radius 3
meters formed by an angle of 45 ̊. Round your
answer to two decimal places.
WARNING! The angle again must be given in radians
Answer:
𝜋
𝜃 = 45° =
4
1 2
𝐴= 𝑟 𝜃
2
1 2
𝐴= 3
2
𝜋
4
r=3m
9𝜋
𝐴=
= 3.53 𝑚2
8
55
Your Turn:
Given an arc of length 4 ft and a circle of radius 7 ft,
find the exact radian measure of the central angle
subtended by the arc; then find the area of the
sector determined by the central angle.
Answer:
𝑠
4
𝑠 = 𝑟𝜃 or 𝜃 = , 𝜃 =
𝑟
7
1 2
1 2
𝐴= 𝑟 𝜃= 7
2
2
4
= 14 𝑓𝑡 2
7
56
Applications
The formula for the length of a circular arc can help you
analyze the motion of a particle moving at a constant speed
along a circular path.
57
Linear and Angular Speed
Linear speed measures how fast the particle moves, and
angular speed measures how fast the angle changes.
r = radius, s = length of the arc traveled, t = time, θ = angle (in radians)
arc length s r
Linear Speed 
 
time
t
t
(distance/time)
Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec
central angle 
Angular Speed 

time
t
(turn/time)
Ex. 6 rev/min, 360°/day, 2π rad/hour
58
Example
A bicycle wheel with a radius of 12
inches is rotating at a constant rate of
3 revolutions every 4 seconds.
a) What is the linear speed of a
point on the rim of this wheel?
12(6 )  18 in /sec
r

t
4
 56.5 in / sec
59
Example
A bicycle wheel with a radius of 12
inches is rotating at a constant rate of
3 revolutions every 4 seconds.
b) What is the angular speed of a
point on the rim of this wheel?
 6 3
 

radians / sec
2
4
t
60
Your Turn
In 17.5 seconds, a car covers an arc
intercepted by a central angle of 120˚
on a circular track with a radius of
300 meters.
a) Find the car’s linear speed in m/sec.
b) Find the car’s angular speed in
radians/sec.
61
Picture it…
300 m
62
Solution
r
(a) Linear speed:
t
Note: θ must be expressed in radians

2
  120 

180
3
 2 
300m 

 3   35.9 m / sec
17.5 sec
63
Solution
(b) Angular speed:  

t
 2 


 3 
17.5 sec
ω ≈ 0.12 radians/sec
64
Example
A race car engine can turn at a maximum rate of 12 000 rpm.
(revolutions per minute).
a)What is the angular velocity in radians per second.
Solution
a) Convert rpm to radians per second
 rev. 
12 000

 min   2  rad  = 1256 radians/s


rev 
 sec 

60 

 min 
65
Your Turn:
The second hand of a clock is 8.4 centimeters long. Find
the linear speed of the tip of this second hand as it passes
around the clock face.
𝑟𝜃
8.4𝑐𝑚 2𝜋
=
= 16.8 𝑐𝑚 𝑚𝑖𝑛
𝑡
1𝑚𝑖𝑛
A lawn roller with a 12-inch radius makes 1.6 revolutions per
second.
Find the angular speed of the roller in radians per second.
Find the speed of the tractor that is pulling the roller.
1.6 𝑟𝑒𝑣 2𝜋 𝑟𝑎𝑑
=
= 3.2𝜋 𝑟𝑎𝑑 𝑠𝑒𝑐
𝑠𝑒𝑐
𝑟𝑒𝑣
𝑟𝜃 (12 𝑖𝑛)(3.2𝜋 𝑟𝑎𝑑)
=
= 38.4𝜋 𝑖𝑛 𝑠𝑒𝑐 = 6.85 𝑚𝑖 ℎ𝑟
𝑡
1 𝑠𝑒𝑐
66
Assignment
Section 4.1, Pg. 265 – 268:Vocab. Check #1 – 10 all
Ex. #1 – 57 odd, 71 – 87 odd, 97, 99
67