Lecture 1.4 Angles & Radian
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Transcript of Lecture 1.4 Angles & Radian
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 1/62
0.1: Angles and RadianMeasure
© 2008 Roy L. Gover (www.mrgover.com)
Learning Goals:
Define radians as anotherangle measure.
Review angle measure
Calculus Prerequisite Knowledge
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 2/62
Definition
Initial Side
Terminal Side
Vertex
Angle Ais instandard
position
A x
y
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 6/62
Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 8/62
Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 9/62
Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 10/62
Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 11/62
Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 12/62
Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 13/62
Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
Is this an angle?
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 14/62
Try This
What kind of angle is this?(hint: whereis theterminal
side?)
quadrantal angle
8/7/2019 Lecture 1.4 Angles & Radian
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Try This What is the
measure of this angle?
a. -90°b. -45°
c. -270°
d. -360°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 16/62
Try This What is the
measure of this angle?
a. -90°b. -45°
c. -270°
d. -360°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 17/62
Try This What is the
measure of thisangle?a. 0°
b. 45°
c. 90°d. 120°
e. 180°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 18/62
Try This What is the
measure of thisangle?a. 0°
b. 45°
c. 90°d. 120°
e. 180°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 19/62
Try This What is the
measure of this angle?
a. -90°b. -45°
c. -270°
d. -360°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 20/62
Try This What is the
measure of this angle?
a. -90°b. -45°
c. -270°
d. -360°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 21/62
Try This
What is themeasure of thisquadrantalangle? x
y
0°
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 22/62
Try ThisIf a 143°angle is in
standard
position,determine
the quadrant in which theterminal side
lies.
x
y
2
8/7/2019 Lecture 1.4 Angles & Radian
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Important Idea
There are two units of measure for angles:
degrees: used in geometryradians: used in calculus
In Precal, we use degreesand radians.
8/7/2019 Lecture 1.4 Angles & Radian
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-1 1
-1
1
y
x
Radian: Thelength of thearc above the
angle dividedby the radius of
the circle.
Definition
sr
U
s
r
U ! , U in radians (rads)
8/7/2019 Lecture 1.4 Angles & Radian
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-1 1
-1
1
y
x
Definition
s
U
1
s
U ! , U in radians (rads)
Unit Circle:the circle withradius of 1
unit If r =1, = sU
1
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
The radian measure of anangle is the distance traveled
around the unit circle. Sincecircumference of a circle is
2 r and r =1, the distancearound the unit circle is 2T T
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Find the degreeand radianmeasure of the
angle in standardposition formed by
rotating the terminal side ½of a circle in the positivedirection. Leave your radian
answer in terms of .T
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Find the degreeand radianmeasure of the
angle in standardposition formed by
rotating the terminal side 5/6of a circle in the negativedirection. Leave your radian
answer in terms of .T
8/7/2019 Lecture 1.4 Angles & Radian
http://slidepdf.com/reader/full/lecture-14-angles-radian 30/62
Try This
Find the degreeand radianmeasure of the
angle in standardposition formed by
rotating the terminal side 2/3of a circle in the positivedirection. Leave your radian
answer in terms of .T
8/7/2019 Lecture 1.4 Angles & Radian
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Solution
2360 240
3r ! rg
2 42
3 3!g radians
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
} rads
360°
rads
6.282T
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Ex ample
45
(degrees)
4
T
radians} .785 radians
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
90
(degrees)
2
T
radians} 1.57 radians
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
} rads
180
radsT
3.14
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Try This
} rads
-180
rads-T
-3.14
8/7/2019 Lecture 1.4 Angles & Radian
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Important Idea
Radian measure allows theexpansion of trig functions
to model real-worldphenomena whereindependent variables
represent distance or timeand not just an angle
measure in degrees.
8/7/2019 Lecture 1.4 Angles & Radian
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Important Idea
If a circle contains 360° or2T radians, how manyradians are in 180°
Use to change
rads to degrees
180°
T rads Use to change
degrees to rads
T rads
180°
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Change 30° to radianmeasure in terms of T .
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
Change 120° to radianmeasure in terms of T .
2 rads3
T
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
Change 240° to radianmeasure in terms of T .
4 rads3
T
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Change radians to
degree measure.
3
4
T
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Change 2.356 radians to
degree measure.
(hint: radians are not always stated in terms of
T .)
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
Change radians to
degree measure.
157.5°
7
8
T
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
Change -3.5 radians to
degree measure to thenearest tenth.
-200.5°
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
0
2
T
T
3
2
T
The
quadrantalangles in
radians
2T
8/7/2019 Lecture 1.4 Angles & Radian
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Definition
0
2
T
T
The
quadrantalangles in
radians
2T
The terminal side is on an
axis.
8/7/2019 Lecture 1.4 Angles & Radian
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DefinitionCoterminal Angles: Anglesthat have the same terminal
side.Important Idea
In precal, angles can belarger than 360° or 2
radians.
T
8/7/2019 Lecture 1.4 Angles & Radian
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Ex ample
Find positive angles andnegative angles that arecoterminal with 30°.
8/7/2019 Lecture 1.4 Angles & Radian
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Important Idea
To find coterminal angles,simply add or subtract
either 360° or 2 radiansto the given angle or any
angle that is alreadycoterminal to the givenangle.
T
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y
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Analysis
30° and1110° havethe same
terminalside,therefore,the anglesare
coterminal
30° x
y
x
y1110°
y
8/7/2019 Lecture 1.4 Angles & Radian
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Analysis
30° and-330° havethe same
terminalside,therefore,the anglesare
coterminal
30° x
y
x
y-330°
8/7/2019 Lecture 1.4 Angles & Radian
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How many angles can youfind that are coterminal
with a 30° angle and howdo you find them?
8/7/2019 Lecture 1.4 Angles & Radian
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Try This
Find 3 angles coterminalwith 60°
420°,780° and -300°
8/7/2019 Lecture 1.4 Angles & Radian
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Try ThisFind one positive angle andtwo negative angle
coterminal with
radians.
32
T
and72
T
2T
52
T ,
8/7/2019 Lecture 1.4 Angles & Radian
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Try ThisFind two positive angle andone negative angle
coterminal with
radians.
56
T
and76
T 196
T 176
T ,