5.1 Angles and Their Measures
Name: _____________
Objectives: Students will be able to convert between radians and
degrees, find arc lengths, convert to nautical miles, and solve problems
involving angular speed.
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
terminal side
θ
vertex
initial side
Standard Position:
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Examples Draw each angle in standard position.
1.) 120o
2.) -90o
3.) 330o
4.) -400o
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Degree: unit of angular measure equal to 1/180th of a straight
angle
In the DMS (degree-minute-second) system of angular measure,
each degree is subdivided into 60 _______ (denoted ') and each
minute is subdivided into 60 ________ (denoted '')
Examples
1.) Convert 37.425o to DMS.
Oct 21­12:34 PM
2.) Convert 42o24'36'' to degrees.
a
a
θ
1 radian
1.) Measure the radius of the circle.
2.) Measure the length around half of
the circle.
3.) How many radians are in a straight
angle?
Jan 11­11:46 AM
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To convert radians to degrees, multiply by ________ .
To convert degrees to radians, multiply by ________ .
Examples
1.) Convert 150o to radians.
2.) Convert 13π/120 to degrees.
Jan 11­11:54 AM
Two positive angles are __________ (or _______________
______) if the sum of their measures is 90o.
Two positive angles are __________ (or _______________
______) if the sum of their measures is 180o.
Examples Find the complement and the supplement of each
angle or explain why the angle has no complement or
supplement.
1.) 73o
2.) 110o
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What is the relationship between
the radius and the degree measure
in radians?
If θ is a central angle in a circle of radius r, and ifθ is
measured in radians, then the length s of the intercepted arc is
given by ________.
Example Find the length of the intercepted arc when r = 7 feet
and θ=35o.
Jan 11­12:05 PM
Area of a Sector
6
in
Example Find the area of a piece of the piece of pizza below.
40o
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Linear and Angular Speed
Suppose an object travels around a circle of radius r. The the
object travels through a central angle of θ radians and an arc
length of s, in time t, then:
1.) v = s/t is the (average) linear speed of the object.
(Unit examples: ft/sec, in/min)
2.) w = θ/t is the (average) angular speed of the object.
(Unit examples: rev/sec, rev/min) Angular speed will be
measured in radians/unit of time. For example: rad/sec
Important Notes: 1 revolution = ____ radians
IMPORTANT
NOTENGULAR
1 radian = ________
Jan 11­12:14 PM
Examples Sammi's truck has wheels 36 inches in diameter. If the
wheels are rotating at 630 rpm, find the truck's speed in mph.
Example A bicycle's wheels are 24 inches in diameter. Assuming
that the bike is travelling at a rate of 25 mph, find the angular
speed of the wheels.
Jan 11­12:08 PM
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Homework: Pages 489-490 #1-59 odd, 61-75 every other odd,
77-89 odd
Jan 11­12:23 PM
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