An Angle of

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Angles of Rotation and Radian Measure
In the last section, we looked at angles that
were acute. In this section, we will look at
angles of rotation whose measure can be
any real number.
An angle of rotation is formed by two rays
with a common endpoint (called the
vertex).
terminal side
initial side
y
The other ray is called the terminal side.
x
One ray is called the initial side.
vertex
Angles of Rotation and Radian Measure
y
terminal side
vertex
initial side
x
The measure of the angle is determined
by the amount and direction of rotation
from the initial side to the terminal side.
The angle measure is positive if the rotation
is counterclockwise, and negative if the
rotation is clockwise.
A full revolution (counterclockwise)
corresponds to 360º.
Angles of Rotation and Radian Measure
This is a positive (counterclockwise) angle
y
x
y
This is a negative
(clockwise) angle
x
Angles of Rotation
That would be a 90º Angle
y
x
y
x
That would be a 180º
Angle
Angles of Rotation
That would be a 270º
Angle
y
x
y
x
That would be a 360º
Angle
Angles of Rotation
An Angle of 120º in
standard position
y
x
y
An Angle of -120º in
standard position
x
Example:
Draw an angle with the given measure in standard
position. Then determine in which quadrant the terminal side
lies.
A. 210º
b. –45º
c. 510º
150º
210º
–45º
Terminal side is
in Quadrant
III
Terminal side is
in Quadrant
IV
510º
Terminal side is
in Quadrant
II
Use the fact that 510º = 360º + 150º.
So the terminal side makes 1 complete
revolution and continues another
150º.
150º
510º
510º and 150º are called coterminal (their
terminal sides coincide).
An angle coterminal with a given angle can be
found by adding or subtracting multiples of
360º.
So if you are asked to find coterminal angles you
can simply add 360 to the angle or subtract 360
from the angle
Find two angles that are coterminal with 130º
(one positive and one negative
130º + 360º = 490º
130º - 360º = -290º
Complimentary and Supplementary Angles
2 angles that are complimentary add up to equal
90 degrees
2 angles that are supplementary add up to equal
180 degrees
Find the supplement to an angle of 24º
180 – 24 = 156
Find the compliment to an angle of 24º
90 – 24 = 66
You can also measure angles in radians.
One radian is the measure of an angle in standard position whose
terminal side intercepts an arc of length r.
r
r
one radian
Since the circumference of a circle is 2πr, there
are 2π radians in a full circle.
Degree measure and radian measure are
therefore related by the following:
360º = 2π radians
Conversion Between Degrees and Radians
• To rewrite a degree measure in radians, multiply by π radians
180º
• To rewrite a radian measure in degrees, multiply by
180º
π radians
Examples:
a.
240º
Rewrite each in radians
b. –90º
4
240º = 240º • π
–90º = –90º • π
180º
180º
3
= –π
= 4π
2
3
c. 135º
3
135º = 135º • π
180º
4
= 3π
4
240º = 4π radians
3
135º = 3π radians
4
–90º = –π radians
2
Examples:
a.
5π
8
Rewrite each in degrees
b.
16π
5
5π = 5π • 180º
8
8
π
= 112.5º
16π = 16π • 180º
5
5
π
= 576º
Two positive angles are complementary if the sum of their measures is π/2 radians
(which is 90º)
Two positive angles are supplementary if the sum of their measures is π radians
(which is 180º).
Example: Find the complement of  = π
The complement is π – π
8
2
8
= 4π – π = 3π
8
8
8
Example: Find the supplement of  = 3π The supplement is π – 3π
5
5
= 5π – 3π 5= 2π
5
5

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