6.1 Angles and Radian Measure
Definitions:
An angle is formed by rotating a ray m, called the initial side of the angle, around its
endpoint until it coincides with a ray n, called the terminal side of the angle. An angle is
made of 2 rays extending from a common endpoint, called the vertex.
A
One of the rays is considered
the initial side while the other
ray is the terminal side.
B
C
< ABC is made of 2 rays,
BA and BC . B is the
common endpoint. B is the
vertex of <ABC. <ABC can
also be named <B (use the
vertex to name the angle).
Negative
angle is
formed.
Positive angle
is formed.
An angle formed by one complete rotation is said to have a measure of 360 degrees.
An angle whose terminal side is in standard position lies along a coordinate axis, the
angle is said to be a quadrantal angle.
QII
360
-360
QIII
QI
QIV
A straight angle measures 180, while a right angle measures 90.
An acute angle measures between 0 and 90.
An obtuse angle measures between 90 and 180.
For two angles to be supplementary, one angle is obtuse and the other is acute or both
angles are right.
For two angles to be complementary, both angles are acute.
Example
Find the degree measure of an angle with 1/9 rotation,
(a) keeping in mind that one complete rotation corresponds to 360.
(b) keeping in mind that one complete rotation corresponds to 2 radians.
Solution:
(a) We are taking 1/9 of a complete rotation. Remembering that the work “of” means to
1 360 360

 40 . So a 1/9 rotation is equivalent to
multiply, we have 1/9 x 360 = 
9
1
9
a 40 angle.
(b) 1/9 x 2 rad.=
1 2 rad . 2 rad . 2



radians . So a 1/9 rotation is equivalent to
9
1
9
9
2/9 radians.
Example:
Find the radian measure of a central angle  opposite a 24 centimeter arc in a circle of
radius 4 centimeters.
Solution:
Looking back to page 417, Definition 2, = s/r radians. In this problem, s is the length of
the arc opposite central angle  and r is the radius of the circle.
s
24cm
  radians 
rad .  6 radians
r
4cm
Example:
Find the exact measure of each angle. Remember that 180= radians.
a.
b.
c.
d.
Change 30 to radian form.
Change 3/4 rad. to degree form.
Change -/3 to degree form.
Change 270 to radian form.
Solutions:
a. 30 
 rad .
180
30  rad . 30  rad . 30 rad . 




 radian
1
180
1
180
180
6

b.
3
180
3 180 540
rad . 
 

 135
4
 rad . 4
1
4
c.

180
 1 180  180
rad . 



 60
3
 rad . 3
1
3
d. 270 
 rad .
180

270  rad . 270  rad . 270
3




rad . 
radians
1
180
1
180
180
2
Example:
Name one positive and one negative angle coterminal with 70º.
Solution:
Recall one complete rotation is 360º. To make an angle more positive, you will add 360º
to the existing angle.
70º + 360º = 430º
To make an angle more negative, you will subtract 360º from the existing angle.
70º - 360º = -290º
NOTE: If by subtracting 360º to the existing angle does not give you a negative angle,
you can continue subtracting 360º as many times needed to achieve a negative angle.
Similar holds true for obtaining positive angles.