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0.1: Angles and RadianMeasure
Learning Goals:
Define radians as anotherangle measure.
Review angle measure
Calculus Prerequisite Knowledge
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Definition
Initial Side
Terminal Side
Vertex
Angle Ais instandard
position
A x
y
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
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Definition
A
If theterminal
sidemovescounter-clockwise,angle A is
positive
x
y
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Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
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Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
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Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
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Definition
A
If theterminal
sidemovesclockwise,angle A isnegative
x
y
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Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
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Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
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Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
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Definition
A
If theterminal
side is onan axis,angle A is aquadrantelangle
x
y
Is this an angle?
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Try This
What kind ofangle is this?(hint: whereis theterminal
side?)
quadrantal angle
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Try This What is the
measure ofthis angle?
a. -90b. -45
c. -270
d. -360
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Try This What is the
measure ofthis angle?
a. -90b. -45
c. -270
d. -360
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Try This What is the
measure of thisangle?a. 0
b. 45
c. 90d. 120
e. 180
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Try This What is the
measure of thisangle?a. 0
b. 45
c. 90d. 120
e. 180
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Try This What is the
measure ofthis angle?
a. -90b. -45
c. -270
d. -360
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Try This What is the
measure ofthis angle?
a. -90b. -45
c. -270
d. -360
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Try This
What is themeasure ofthisquadrantalangle? x
y
0
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Try ThisIf a 143angle is in
standard
position,determine
the quadrantin which theterminal side
lies.
x
y
2
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Try ThisIf a 280angle is in
standard
position,determine
the quadrantin which theterminal side
lies.
x
y
4
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Important Idea
There are two units ofmeasure for angles:
degrees: used in geometryradians: used in calculus
In Precal, we use degreesand radians.
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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)
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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
unitIfr=1, =sU
1
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Definition
The radian measure of anangle is the distance traveled
around the unit circle. Sincecircumference of a circle is
2 rand r=1, the distancearound the unit circle is 2T T
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Example
Find the degreeand radianmeasure of the
angle in standardposition formed by
rotating the terminal side of a circle in the positivedirection. Leave your radian
answer in terms of .T
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Example
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
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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
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Solution
2360 240
3r ! rg
2 42
3 3!g radians
X
X
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Example
} rads
360
rads
6.282T
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Example
45
(degrees)
4
T
radians} .785 radians
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Example
90
(degrees)
2
T
radians} 1.57 radians
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Try This
} rads
180
radsT
3.14
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Try This
} rads
-180
rads-T
-3.14
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Try This
} rads
270
rads
4.71
1 3
1 or2 2
T
T
Do you see a pattern?
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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.
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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
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Example
Change 30 to radianmeasure in terms ofT.
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Try This
Change 120 to radianmeasure in terms ofT.
2 rads3
T
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Try This
Change 240 to radianmeasure in terms ofT.
4 rads3
T
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Example
Change radians to
degree measure.
3
4
T
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Example
Change 2.356 radians to
degree measure.
(hint: radians are notalways stated in terms of
T.)
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Try This
Change radians to
degree measure.
157.5
7
8
T
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Try This
Change -3.5 radians to
degree measure to thenearest tenth.
-200.5
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Definition
0
2
T
T
3
2
T
The
quadrantalangles in
radians
2T
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Definition
0
2
T
T
3
2
T
The
quadrantalangles in
radians
2T
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Definition
0
2
T
T
3
2
T
The
quadrantalangles in
radians
2T
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Definition
0
2
T
T
The
quadrantalangles in
radians
2T
The terminal side is on an
axis.
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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
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Example
Find positive angles andnegative angles that arecoterminal with 30.
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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
y
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Analysis
30 and390 havethe same
terminalside,therefore,the anglesare
coterminal
30x
y
x
y
390
y
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Analysis
30 and750 havethe same
terminalside,therefore,the anglesare
coterminal
30x
y
x
y
750
y
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Analysis
30 and1110 havethe same
terminalside,therefore,the anglesare
coterminal
30x
y
x
y1110
y
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Analysis
30 and-330 havethe same
terminalside,therefore,the anglesare
coterminal
30x
y
x
y-330
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How many angles can youfind that are coterminal
with a 30 angle and howdo you find them?
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Try This
Find 3 angles coterminalwith 60
420,780 and -300
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Try ThisFind one positive angle andtwo negative angle
coterminal with
radians.
32
T
and72
T
2T
52
T,
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Try ThisFind two positive angle andone negative angle
coterminal with
radians.
56
T
and76
T 196
T 176
T,
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Lesson CloseIn your own words andwithout looking at your
notes, write a definition for:
Coterminal angle
Radian
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