aat-a 4/25/14 obj: swbat convert from degrees to radians and vice versa. agenda bell ringer:...
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AAT-A 4/25/14
Obj: SWBAT convert from degrees to radians and vice versa.Agenda•Bell Ringer: Inquiry: Angle measure•HW Requests: Comments on ACT•Turn in Break Packets•Tabled: Skills Practice WS Angle of Elevation and Depression•Chapter 8 Take Home Test•Homework: Read Section 13.2 #19-49 odds
Announcements:
Larie, Midterm???
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6.1 Radian and Degree Measure
In this section, we will study the following topics:
Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree measure Find coterminal angles
Angle-formed by rotating a ray about its endpoint (vertex)
Initial Side Starting position
Terminal Side Ending position
Standard Position Initial side on positive x-axis and the vertex is on the origin
An angle describes the amount and direction of rotation
120° –210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
When sketching angles, always use an arrow to show direction.
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6.1 Radian and Degree Measure
Measuring Angles
The measure of an angle is determined by the amount of
rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees
and in radians.
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of of one
revolution.
1
360
6
6.1 Radian and Degree Measure
Measuring Angles
1
360
7
In general, for in radians,
A second way to measure angles is in radians.
Radian Measure
s
r
Definition of Radian:
One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.
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Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 902
2 6.28
3.14
1.572
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Conversions Between Degrees and Radians
1. To convert degrees to radians, multiply degrees by
2. To convert radians to degrees, multiply radians by
180
180
Example
Convert from degrees to radians: 210º
210
Convert from degrees to radians.1. 542. -300
Convert from radians to degrees.3.
4.
11
3
13
12
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6.1 Radian and Degree Measure
Coterminal Angles
Angles that have the same initial and terminal sides are
coterminal.
Angles and are coterminal.
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6.1 Radian and Degree Measure
Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle by
adding or subtracting multiples of 360º.
Ex 2:
Find one positive and one negative angle that are
coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º
Coterminal Angles:Two angles with the same initial
and terminal sides
Find a positive coterminal angle to 20º 38036020
34036020Find a negative coterminal angle to 20º
Types of questions you will be asked:
Identify a) ALL angles coterminal with 45º, then b) find one positive coterminal angle and one negative coterminal angle.
a) 45º + 360k (where k is any given integer).
b) Some possible answers are 405º, 765º, - 315º, - 675º
To find a coterminal angle add or subtract multiples of 360º
if in degrees or 2π if in radians
Ex 5. Convert the degrees to radian measure.
a) 60
b) 30 c) d) -54
e) -118f)
g) 45
Class Work
a)
b)
c)
d)
6
2
11
18
9
15
Class Work Time permittingPg 712 #1, 2, 4-16
HW: Read Section 13.2 #19-49 odds
Ex 6. Convert the radians to degrees.
a)
b)
c)
d)
6
2
11
18
9
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Angles are often classified according to the quadrant
in which their terminal sides lie.
Ex1: Name the quadrant in which each angle lies.
50º
208º II I
-75º III IV
6.1 Radian and Degree Measure
Classifying Angles
Quadrant 1
Quadrant 3
Quadrant 4
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6.1 Radian and Degree Measure
Classifying Angles
Standard position angles that have their terminal side
on one of the axes are called quadrantal angles.
For example, 0º, 90º, 180º, 270º, 360º, … are
quadrantal angles.
Ex 3. Find one positive and one negative angle that is coterminal with the angle = 30° in standard position.
Ex 4. Find one positive and one negative angle that is coterminal with the angle = 272 in standard position.
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6.1 Radian and Degree Measure
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.
s
r
In general,
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6.1 Radian and Degree Measure
Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 902
2 6.28
3.14
1.572
22
6.1 Radian and Degree Measure
Radian Measure
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6.1 Radian and Degree Measure
Conversions Between Degrees and Radians
1. To convert degrees to radians, multiply degrees by
2. To convert radians to degrees, multiply radians by
180
180
Ex 7. Find one positive and one negative angle that is coterminal with the angle = in standard position.
Ex 8. Find one positive and one negative angle that is coterminal with the angle = in standard position.7
5
3
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0°
360 °
30 °
45 °
60 °
330 °
315 °
300 °
120 °
135 °
150 °
240 °
225 °
210 °
180 °
90 °
270 °
Degree and Radian Form of “Special” Angles
Find one postive angle and one negative angle in standard position that are coterminal with the given angle.
5. 135
6. 11
6
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