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RADIAN - DEGREE
A radian is the angle measure of a perfectly cut piece of pie.
r
r
r
As soon as we see
that the arc length
equals the radius,
we know the central
angle is 1 radian.
Since circumference C of
a circle with radius r is
calculated using 2 r,
It follows that a complete
rotation will produce an
angle of 2 radians
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n rad180 rad
=
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RADIAN MEASURE
When we construct a circle graph, we assume that the area of a sectorof a
circle is proportional to the sector angle.
The length of the arc bounding the sector is proportional to the sector angle and is
called the arc length.
O
Arc Length
Sector Angle
Sector
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Arc Length of Sector Sector Angle
Circumference Full-turn Angle=
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EX 1:
Calculate the arc
length of a sector of a
circle of radius 20 cmif the sector angle is
140.
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EX 1: CALCULATE THE ARC LENGTH OF A
SECTOR OF A CIRCLE OF RADIUS 20 CM
IF THE SECTOR ANGLE IS 140.
Solution Create a visual
What do we know? r = 20 cm angle = 140
L = 2r 360
L = 2r L = (140)(2) (20) = 49cm
360 360
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EX 2:
Find the measure of theangle, to the nearest
tenth of a degree,
subtended at the center
of a circle, radius R, by
the arc of each length
a) R b) 2R c) 3R
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SOLUTIONS
Go back to the original proportion:
Arc Length = Sector Angle
Circumference Full-turn Angle
What do we know?
a) R = = 360 = 57.3
2R 360 2
b) 2R = = 360 = 114.6
2R 360
c) 3R = = (3)(360) = 171.9
2R 360 2
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In Example 2 we discovered that an angle of 180/ {approximately 57} is subtended at
the center of a circle by an arc length of R, where R is the radius.
Definition: One RADIAN is the measure of an angle which is subtended at the center of acircle by an arc equal in length to the radius of the circle.
From this definition: 1 radian = 180/
Multiply both sides by to get the following results:
radians = 180
Therefore, a full-tern angle, 360 is equal to 2 radians.
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DOES THIS LOOK FAMILIAR?
Arc Length of Sector Sector Angle
Circumference Full-turn Angle=
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=
In DEGREES
r180
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=
In RADIANS
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EX 3:
A circle has radius 6.5 cm.Calculate the length of an
arc of this circle
subtended by each angle.
a) 2.4 radians b) 75
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SOLUTION
a) L = r= (6.5)(2.4)
= 15.6 cm
b) L = r/180
= (75) (6.5)/180
= 8.5 cm
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QUESTIONS
1. Convert from degrees to radians. Express the answer interms of .
a) 30 b) 60 c) 225 d) 300
e) 330 f) 405 g) 120 h) 270
2. Covert from radians to degrees.
a) /2rads b) -2 /3rads c) 3/4rads d)2rads
e) -3/2rads f) 7/4rads g) -11/6rads h) /6rads
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3. Find the length of the arc which subtends each angle at the
center of a circle of radius 5cm. Give answers to 1 decimal place.
a) 2.0 rads b) 3.0 rads c) 1.8 radsd) 6.1 rads e) 4.2 rads f) 0.6 rads
4. Find the length of the arc of a circle with radius 12 cm that
subtents each sector angle. Give answers to 1 decimal place wherenecessary
a) 135 b) 75 c) 105 d) 165
e) 240 f) 180 g) 310 h) 200
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5. Find the arc length to the nearest centimetre of the sector
of a circle with radius:
a) 7m, if the sector angle is:
i) 120 ii) 210
b) 90cm, if the sector angle is:
i) 30 ii) 225c) 216mm, if the sector angle is:
i) 135 ii) 300
6. How many radians are there in:
a) A full turn b) a half turn c) a quarter turn
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7. Calculate the arc length to the nearest metre of a
sector of a circle with radius 6m if the sector
angle is 140
8. Two sectors of the same circle have sector angles
of 35 and 105 respectively. The arc length ofthe smaller sector is 17cm. What is the arc
length of the larger sector?
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9. The Earth travels in a nearly circular orbit around the sun.
The radius of the orbit is about 149 000 000 km.
a) What is the measure in radians of the angle subtended atthe sun by the position of the Earth at two different times
24h apart?
b) About how far does the Earth travel in one day in its orbit
around the sun?Earth now
Earth 24 hrs later
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10. The angular velocityof an object is the angle per unit
time through which an object rotates about a rotation
center.
a) What is the angular velocity in radians per second of a
car tire of diameter 64cm when the car is travelling at
100km/h?
b) Write an expression for the angular velocity in radians
per second for a car tire of diameter d centimeters
when the car is travelling at x kilmeters per hour.
1 Centimeter per Second = 0.036 Kilometers per Hour
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QUESTIONS
1. Convert from degrees to radians. Express the answer in terms of .
a) 30 b) 60 c) 225 d) 300
e) 330 f) 405 g) 120 h) 270
2. Covert from radians to degrees.
a) /2 b) -2 /3 c) 3/4 d)2
e) -3/2 f) 7/4 g) -11/6 h) /6
/6 /3 5/4 5/3
11/6 9/4 2/3 3/2
90 120 135 360
-270 315 - 330 30
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3. Find the length of the arc which subtends each angle at the
center of a circle of radius 5cm. Give answers to 1 decimal place.
a) 2.0 rads b) 3.0 rads c) 1.8 rads
d) 6.1 rads e) 4.2 rads f) 0.6 rads
4. Find the length of the arc of a circle with radius 12 cm that
subtents each sector angle. Give answers to 1 decimal place wherenecessary
a) 135 b) 75 c) 105 d) 165
e) 240 f) 180 g) 310 h) 200
10 cm 15cm 9cm
30.5cm 21cm 3cm
28.3cm 15.7cm 22cm 34.5cm
50.2cm 37.7cm 64.9cm 41.9cm
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5. Find the arc length to the nearest centimetre of the sector
of a circle with radius:
a) 7m, if the sector angle is:
i) 120 ii) 210
b) 90cm, if the sector angle is:
i) 30 ii) 225c) 216mm, if the sector angle is:
i) 135 ii) 300
6. How many radians are there in:
a) A full turn b) a half turn c) a quarter turn
14.7 m 25.6 m
47.1cm 353.3 cm
508.7mm 1130.4mm
2 /2
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7. Calculate the arc length to the nearest meter of a
sector of a circle with radius 6m if the sector
angle is 140
8. Two sectors of the same circle have sector angles of 35
and 105 respectively. The arc length of the smaller sectoris 17cm. What is the arc length of the larger sector?
140 6 180 = 14.7m
Radius small circle = radius big circle
LS = 17 = 35 r 180 r = 27.8435
Using found r and LB equation LB = 51cm
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9. The Earth travels in a nearly circular orbit around the sun.
The radius of the orbit is about 149 000 000 km.
a) About how far does the Earth travel in one day in its orbit around the sun?
b) What is the measure in radians of the angle subtended at the sun by the position
of the Earth at two different times 24h apart? Earth now
Earth 24 hrs later
The Earth circles the sun once a year, how far does it go?
FIND CIRCUMFERENCE: 935720000km per year 365 days
Therefore it travels 2563616.43 km per day
= L/r = 00.0172 rads
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10. The angular velocityof an object is the angle per unit
time through which an object rotates about a rotation
center.a) What is the angular velocity in radians per second of a car tire of diameter 64cm
when the car is travelling at 100km/h?
1 Centimeter per Second = 0.036 Kilometers per Hour
Angular Velocity = 360/t (seconds) = 2/t(seconds)
Ctire = 200.96
How many rotations per second?
100km/hr = 2777.7cm/s Vang= 360 2
0.0723476 0.07
2777.7 cm = 200.96 = 4975/s = 86.8 rads
s ?
= 0.0723476 seconds per rotation
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Write an expression for the angular velocity in radians per second for a car tire of
diameter d centimeters when the car is travelling at x kilometers per hour.
Vang= 360 = 2 where t is measured in seconds
t t
Speed (cm/s) = Circumference time = Circumference = 2r
time Speed y
y (cm/s) = 0.036x (km/h)
we need d not r
Vang= 2
d 2 1
0.036x d 0.036X
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