Radian and Degree Measure - Social Circle City Schools
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Transcript Radian and Degree Measure - Social Circle City Schools
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
WHY???
You can use angles to
model and solve real-life
problems. For example,
you can use angles to find
the speed of a bicycle
ANGLES:
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
An angle is formed by joining the endpoints
of two half-lines called rays.
The side you measure to is called the terminal side.
Angles measured counterclockwise are given a
positive sign and angles measured clockwise are
given a negative sign.
Negative Angle
This is a clockwise
rotation.
Positive Angle
This is a
counterclockwise
rotation.
Initial Side
The side you measure from is called the initial side.
Angle describes the amount and direction of
rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
What is the measure of this angle?
You could measure in the positive
= - 360 + 45° direction and go around another rotation
which would be another 360°
= - 315°
= 45°
You could measure in the positive
direction
= 360 + 45°= 405°
You could measure in the negative
direction
There are many ways to express the given angle.
Whichever way you express it, it is still a Quadrant I
angle since the terminal side is in Quadrant I.
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
Radian = the measure of a central angle θ that intercepts
an arc “s” equal in length to the radius “r” of the circle
Because the circumference of a circle is 2πr units, a central angle of
one full rotation (counterclockwise) corresponds to an arc length of s =
2πr.
The radian measure of a central angle θ is obtained by dividing the arc
length s by r (θ = s/r)
Another way to measure angles is using what is called
radians.
Given a circle of radius r with the vertex of an angle as the
center of the circle, if the arc length formed by intercepting
the circle with the sides of the angle is the same length as
the radius r, the angle measures one radian.
r
r
r
radius of circle is r
arc length is
also r
initial side
This angle measures 1
radian
2
QUADRANT II
2
QUADRANT I
0
2
0
QUADRANT III
QUADRANT IV
3
2
3
2
3
2
2
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 positive angles
subtract 2π and for negative angles
add 2π.
Coterminal
Angles:
Two angles with the same
initial and terminal sides
Find a positive coterminal angle to 20º
20 360 380
Find a negative coterminal angle to 20º
15
Find 2 coterminal angles to
4
20 360 340
23
15 8
15
2
4
4
4
4
15
15 8 7 8
2
4
4
4
4
4
4
COTERMINAL ANGLES EXAMPLE
Find two Coterminal Angles (+ and -)
13
6
3
4
2
3
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
3 2
2
3
3
3
3
COMPLEMENTARY & SUPPLEMENTARY ANGLES
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
DEGREE MEASURE
Since 2π radians corresponds to one complete
revolution, degrees and radians are related by
the equations. . .
360° = 2π rad
1
180
rad
180° = π rad
1rad
180
CONVERSIONS BETWEEN DEGREES
AND RADIANS
To apply these two conversion rules, use the basic
relationship π = 180°
To
convert degrees to radians,
multiply degrees by
180
To
convert radians to degrees,
multiply radians by 180
Convert to radians:
135
180
3
4
Convert to degrees:
8 180
480
3
5 180
6
150
80
180
4
9
EXAMPLES:
Convert from degrees to radians
135°
540°
-270°
Convert from radians to degrees
2
rad
9
rad
2
2rad
A Sense of Angle Sizes
45
4
30
6
90
2
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.
Degrees – Minutes - Seconds
With calculators it is easy to denote fractional parts of degrees in
decimal form. Historically, however, fractional parts of degrees were
expressed in minutes and seconds by using prime (‘) and double
prime (“) notations. The graphing calculator can also aid in these
conversions.
ANGLE Menu
To display the ANGLE menu, press 2nd (ANGLE). The ANGLE menu
displays angle indicators and instructions. The Radian/Degree mode
setting affects the TI-83 Plus’s interpretation of ANGLE menu entries.
Option 4 - DMS (degree/minute/second)
This displays answer in DMS format. The mode setting must be
Degree for answer to be interpreted as degrees, minutes and
seconds. DMS is valid only at the end of a line.
1 degree = 60 minutes
1 minute = 60 seconds
1° = 60
1 = 60
So … 1 degree = _________seconds
3600
Express 365010as decimal degrees
Express
50.525 in degrees, minutes, seconds
50º + .525(60)
50º + 36.5
50º + 36 + .5(60)
50 degrees, 36 minutes, 30
seconds
Arc length s of a circle is found with the following formula:
s = r
arc length
radius
IMPORTANT: ANGLE MEASURE MUST
BE IN RADIANS TO USE FORMULA!
measure of angle
Find the arc length if we have a circle with a radius of 3
meters and central angle of 0.52 radian.
arc length to find is in black
= 0.52
3
s = 3r0.52
= 1.56 m
What if we have the measure of the angle in degrees? First you must convert
from degrees to radians before you use the formula.
ARC LENGTH
s = rθ (s = arc length, r = radius, θ= central
angle measure in radians)
EXAMPLES:
A circle has a radius of 4 inches. Find the length of
the arc intercepted by a central angle of 240°.
A circle has a radius of 8 cm. Find the length of the
arc intercepted by a central angle of 45°
LINEAR AND ANGULAR SPEED
Linear speed measure 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
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?
r 12(6 )
18 in / sec
t
4
56.5 in / sec
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
t
2
4
Example 2
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.
Picture it…
300 m
Example 2
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
Example 2
(b) Angular speed:
t
2
3
17.5 sec
ω ≈ 0.12 radians/sec
Example 3
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
EXAMPLES:
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.
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.