§5.1 Angles and Their Measure Objectives 1. Convert...
Transcript of §5.1 Angles and Their Measure Objectives 1. Convert...
§5.1 Angles and Their Measure
Objectives
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1. Convert between decimal degrees and degrees, minutes,
seconds measures of angles.
2. Find the length of an arc of a circle.
3. Convert from degrees to radians and from radians to degrees.
4. Find the area of a sector of a circle.
5. Find the linear speed of an object travelling in circular
motion.
§5.1 Angles and Their Measure
Ray & Vertex
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A ray ( or half-line) is that portion of a line that starts at a point V on
the line and extends indefinitely in one direction. The starting point V
of the ray is called its vertex.
LineRay
V
§5.1 Angles and Their Measure
Angle, Initial & Terminal Side, Positive & Negative Angles
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An angle is formed by two rays that share a common vertex. The angle
is formed by rotating from the initial side to the terminal side. If the
rotation is counterclockwise, the angle is positive. If the rotation is
clockwise, the angle is negative.
V Initial Side
Positive
Angle, a
a
V Initial Side V Initial Side
b
Negative
Angle, b
g
Positive
Angle, g
N.B.: a ≠ b ≠ g but a, b and g are said to be co-terminal!
§5.1 Angles and Their Measure
Angle, Initial & Terminal Side, Positive & Negative Angles
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Definition from Supplemental Packet: The reference angle for an
angle in standard position is the positive acute angle formed by the
horizontal axis and the terminal side of the angle.
V Initial Side
Reference
Angle, a
a
V Initial Side V Initial Side
b
Reference
Angle, a
g
Reference
Angle, a
a a
N.B.: a ≠ b ≠ g but a, b and g are said to be co-terminal!
§5.1 Angles and Their Measure
Standard Position
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a) q in standard position;
q > 0
b) q in standard position;
q < 0
q
Initial side
Terminal side
Vertex x
y
q
Initial side
Terminal side
Vertexx
y
§5.1 Angles and Their Measure
Drawing an Angle
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Draw each angle:
a) a = 45º b) b =-90º c) q = 225º d) f = 405º
§5.1 Angles and Their Measure
Drawing an Angle
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Draw each angle:
a) a = 45º b) b =-90º c) q = 225º d) f = 405º
§5.1 Angles and Their Measure
1: Conversions Decimal Degrees and DºM’S”
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1 counterclockwise revolution = 360º
1º = 60 minutes = 60’
1’ = 60 seconds = 60”
Example computation:
a) Convert q = 45º 10’15” to decimal degrees. Round the answer to
three decimal places.
§5.1 Angles and Their Measure
1: Conversions Decimal Degrees and DºM’S”
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1 counterclockwise revolution = 360º
1º = 60 minutes = 60’
1’ = 60 seconds = 60”
Example computation:
b) Convert q = 21.56º to DºM’S”. Round the answer to the nearest
second.
§5.1 Angles and Their Measure
1: Conversions Decimal Degrees and DºM’S”
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Example computation:
a) Convert q = 45º 10’15” to decimal degrees. Round the answer to
three decimal places.
b) Convert q = 21.56º to DºM’S”. Round the answer to the nearest
second.
1º 1' 1º45º10 '15'' 45º 10 ' 15'' 45.171º
60 ' 60 '' 60 '
q
60 ' 60 ''21.56º 21º 0.56º 21º 33.6'=21º33'+0.6'
1º 1'
21º33'36 ''
q
1 counterclockwise revolution = 360º
1º = 60 minutes = 60’
1’ = 60 seconds = 60”
§5.1 Angles and Their Measure
Angles in Radians
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A central angle is a positive angle whose vertex is at the centre of a
circle of radius r. The rays of a central angle subtend (intersect) an arc
on the circle. If the arc length is s, then the angle in radians is:
If the arc length is s = r, then q = 1 radian.
arc length
radius
s
rq
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§5.1 Angles and Their Measure
2: The Length of an Arc of a Circle
Angle in radians:
arc length
radius
s
rq
(r, 0)
Initial Side
1 1s r q
1q
2 2s r q
2q 3q
3 3s r q
21 revolution 2
r
r
N.B.:
• An angle in radians is
length over length,
hence a dimensionless
quantity.
• For a circle of radius r, a
central angle q in
radians subtends an arc
of length s such that:
s = r q
§5.1 Angles and Their Measure
3: Unit Conversions
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1 revolution = 2 radians = 360º ⇒ radians = 180º
1degree radian180
1801radian degree
Example: Convert each angle in degrees to radians:
(a) 80° (b) 140° (c) -30° (d) 100°
§5.1 Angles and Their Measure
3: Unit Conversions
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1 revolution = 2 radians = 360º ⇒ radians = 180º
1degree radian180
1801radian degree
Example: Convert each angle in degrees to radians:
(a) 80° (b) 140° (c) -30° (d) 100°
rad 4
(a) 80º rad180º 9
rad 1
(c) 30º rad180º 6
§5.1 Angles and Their Measure
3: Unit Conversions
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Example: Convert each angle in radians to degrees:
(a) 2/3 (b) 5/6 (c) 3/5 (d) 2
§5.1 Angles and Their Measure
3: Unit Conversions
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Example: Convert each angle in radians to degrees:
(a) 2/3 (b) 5/6 (c) 3/5 (d) 2
2 180º(a) 120º
3 rad
5 180º(b) 150º
6 rad
3 180º(c) 108º
5 rad
180º 360º(d) 2
rad
Degrees 0º 30º 45º 60º 90º 120º 135º 150º 180º
Radians 0 /6 /4 /3 /2 2/3 3/4 5/6
Degrees 210º 225º 240º 270º 300º 315º 330º 360º
Radians 7/6 5/4 4/3 3/2 5/3 7/4 11/6 2
§5.1 Angles and Their Measure
3: Unit Conversions
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§5.1 Angles and Their Measure
Example: Finding the Distance Between Two Cities
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The latitude of a location L is the angle formed by a ray drawn from the
centre of Earth to the Equator and a ray drawn from the centre of Earth to
L. See Figure. Glasgow, Montana, is due north of Albuquerque, New Mexico. Find the distance between Glasgow (48º9’ north latitude) and
Albuquerque (35º5’ north latitude). See Figure 13(b). Assume that the
radius of Earth is 3960 miles.
§5.1 Angles and Their Measure
Example: Finding the Distance Between Two Cities
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Given:
r = 3960 miles
qA = 35º 5’
qA
qG
q
s
r
Let:
13º4 '
rad 1º rad13º 4 '
180º 60 ' 180º
G Aq
q
q
rad 1º rad
3960 miles 13º 4 '180º 60 ' 180º
903.10miles
s r q
qG = 48º 9’
Glasgow
Albuquerque
§5.1 Angles and Their Measure
4: Area of a Sector
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1 1
In general:
A
A
q
q
For q1 = 2 ⇒ A1 = r 2
221
1
1
2 2
A rA r
q q q
q
The area A of the sector of a circle of radius r formed by a central
angle q radians is:
21
2A r q
§5.1 Angles and Their Measure
Example: Area of a Sector
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Find the area of the sector of a circle of radius 5 feet formed by an
angle of 40°. Round the answer to two decimal places.
§5.1 Angles and Their Measure
Example: Area of a Sector
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Find the area of the sector of a circle of radius 5 feet formed by an
angle of 40°. Round the answer to two decimal places.
2
2
2
2
1
2
15ft 40º
2 180º
25ft
9
8.73 ft
A r q
Method 1: Recipe
§5.1 Angles and Their Measure
Example: Area of a Sector
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Method 2: Conceptual
Find the area of the sector of a circle of radius 5 feet formed by an
angle of 40°. Round the answer to two decimal places.
2
2
2
2
40
360
15ft
9
25ft
9
8.73 ft
A r
§5.1 Angles and Their Measure
5: Linear & Angular Speed for Circular Motion
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position at
time: t = 0
position at
time: t > 0
q
s v t
linear speedv
s
t
angular speed
t
q
v r
§5.1 Angles and Their Measure
Example: Finding Linear Speed
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A child is spinning a rock at the end of a 2-foot rope
at the rate of 180 revolutions per minute (rpm). Find
the linear speed of the rock when it is released.
rev rad2[ft] 180 2
min rev
ft720
min
ft2261.95
min
v r
r
v r
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§5.1 Angles and Their Measure
Linear Speed and Angular Speed
Two insects sit on an old vinyl record. A red insect sits at a point that is 1
inch from the centre of the disk and a green insect sits at a point that is 6
inches from the centre. The record rotates such that it completes 331/3
revolutions per minute. Present the analysis to determine:
a) the exact value of the angular speed of each insect in radian/sec,
b) the linear speed of each insect (rounded to the nearest inch/sec) , and
c) the total distance (rounded to the nearest inch) travelled by each
insect after 3 minutes.
§5.1 Angles and Their Measure
Linear Speed and Angular Speed
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Two insects sit on an old vinyl record. A red insect
sits at a point that is 1 inch from the centre of the
disk and a green insect sits at a point that is 6 inches
from the centre. The record rotates such that it
completes 331/3 revolutions per minute. Present the
analysis to determine:
a) the exact value of the angular speed of each
insect in radian/sec,
b) the linear speed of each insect (rounded to the nearest inch/sec) , and
§5.1 Angles and Their Measure
Linear Speed and Angular Speed
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c) the total distance (rounded to the nearest inch)
travelled by each insect after 3 minutes.
Two insects sit on an old vinyl record. A red insect
sits at a point that is 1 inch from the centre of the
disk and a green insect sits at a point that is 6 inches
from the centre. The record rotates such that it
completes 331/3 revolutions per minute. Present the
analysis to determine:
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§5.1 Angles and Their Measure
Linear Speed and Angular Speed
The minute hand of a clock is 6 centimetres long. You are very bored by
the lecture, so you watch it rotate for 35 minutes. Present the analysis to
exact values for:
a) The angle in degrees spanned by
the minute hand after 35 minutes.
b) The angle in radians spanned by the
minute hand after 35 minutes.
c) The distance travelled by the tip of
the minute hand after 35 minutes.
d) The linear speed of the tip of the
minute hand in centimetres per
minute.
e) The angular speed of the tip of the
minute hand in radians per minute6
39
1
2
4
57
8
10
1112
§5.1 Angles and Their Measure
Linear Speed and Angular Speed
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a) The angle in degrees spanned by the minute hand after 35 minutes.
b) The angle in radians spanned by the minute hand after 35 minutes.
c) The distance travelled by the tip of the minute hand after 35
minutes.
d) The linear speed of the tip of the minute hand in centimetres per
minute.
e) The angular speed of the tip of the minute hand in radians per
minute
§5.1 Angles and Their Measure
Example: Finding Linear Speed
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Earth rotates on an axis through its poles. The distance from the
axis to a location on Earth 40° north latitude is about 3033.5 miles.
Therefore, a location on Earth at 40° north latitude is spinning on a
circle of radius 3033.5 miles. Compute the linear speed on the
surface of Earth at 40° north latitude.
distance travelled
elapsed time
2
2 3033.5miles
24hour
794.17 miles/hour
sv
t
r
T
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§5.1 Angles and Their Measure
Angles in Radians – Another Example Computation
A weather satellite orbits the Earth in a circular
orbit 500 miles above the Earth's surface.
Assume Earth to be a perfect sphere with
radius 3960 miles and the satellite has
travelled a distance of 600 miles. Present the
analysis to find:
a) exact values for the satellite's angular
displacement in radians and in degrees of arc and
b) approximate values for the satellite's angular displacement rounded to
the nearest tenth of a radian and to the nearest tenth of a degree.