### 9.2 Define General Angles and Use Radian Measure

```9.2 Define General Angles and
What are angles in standard position?
Angles in Standard Position
In a coordinate plane, an angle
90°
can be formed by fixing one ray
called the initial side and
rotating the other ray called the 180°
vertex
An angle is in standard position
if its vertex is at the origin and
270°
its initial side lies on the positive
x-axis.
The measure of an angle is positive if the rotation of its
terminal side is counterclockwise and negative if the
rotation is clockwise.
The terminal side of an angle can make more than one
complete rotation.
0°
Draw an angle with the given measure in standard
position.
a. 240º
SOLUTION
a.
Because 240º is 60º more
than 180º, the terminal side
is 60º counterclockwise past
the negative x-axis.
Draw an angle with the given measure in standard
position.
b.
500º
SOLUTION
b.
Because 500º is 140º more
than 360º, the terminal side
makes one whole revolution
counterclockwise plus 140º
more.
Draw an angle with the given measure in standard
position.
c.
–50º
SOLUTION
c.
Because –50º is negative, the
terminal side is 50º clockwise
from the positive x-axis.
Coterminal Angles
Coterminal angles are angles whose terminal sides
coincide.
An angle coterminal with a given angle can be found
by adding or subtracting multiples of 360°
The angles 500° and 140°
are coterminal because their
terminal sides coincide.
Find one positive angle and one negative angle that are
coterminal with (a) –45º
SOLUTION
There are many such angles, depending on what
multiple of 360º is added or subtracted.
a.
–45º + 360º = 315º
–45º – 360º = – 405º
Find one positive angle and one negative angle that are
coterminal with (b) 395º.
b. 395º – 360º = 35º
395º – 2(360º) = –325º
Draw an angle with the given measure in standard
position. Then find one positive coterminal angle and
one negative coterminal angle.
1.
65°
65º + 360º = 425º
65º – 360º = –295º
2.
230°
230º + 360º = 590º
230º – 360º = –130º
3.
300°
300º + 360º = 660º
300º –
360º
4.
= –60º
740°
740º –
2(360º)
740º –
3(360º)
= 20º
= –340º
One radian is the measure of an
angle in standard position whose
terminal side intercepts an arc of
length r.
Because the circumference of a
circle is 2, there are 2 radians
in a full circle.
measure are related by the
equation 360° = 2 radians or
Multiply degree measure
180°
measure by
by

180°

Degree and Radian Measures of Special Angles
The diagram shows equivalent
special angles for 0° to 360°
It will be helpful to memorize
the equivalent degree and
angles in the 1st quadrant and

2
All other special angles are
multiples of these angles.
12
degrees.
a. 125º = 125º
180º
=
25π
36
(
)(
)
π
180º
π
–
b. –
=
12
12
= –15º
measure to degrees.
5. 135°
135º = 135º
=
(
3π
4
180º
(
–50° = –50°
6. –50°
=
7.
5π
4
(
5π
4 =
– 5π
18
180º
)(
)
5π
180º
4
= 225º
8.
π
10
π
10
=
(
= 18º
π
10
)(
)
180º
Sectors of Circles
A sector is a region of a circle
that is bounded by two radii
and an arc of the circle.
The central angle  of a sector
is the angle formed by the two
Arc Length and Area of a Sector
The arc length s and
area A of a sector with
angle  (measures in
Arc length:  =
Area:  =

Softball A softball field forms a sector with the dimensions
shown. Find the length of the outfield fence and the
area of the field.
SOLUTION
STEP 1 Convert the measure of
the central angle to
π
90º = 90º
=
2
180º
STEP 2 Find the arc length and the area of the sector.
Arc length: s = r θ= 180 ( π )= 90π ≈ 283 feet
2
1
Area: A = 1 r2θ =
(180)2 ( π ) = 8100π ≈ 25,400 ft2
2
2
2
(
Arc length: s = r θ= 180 ( π )= 90π ≈ 283 feet
2
1
Area: A = 1 r2θ =
(180)2 ( π ) = 8100π ≈ 25,400 ft2
2
2
2