Pre – Calculus Math 12
Angles & Angle Measure
Lesson Focus: To convert angles in degree measure to radian measure and vice versa; to determine the
measures of angles that are co-terminal with a given angle; to solve problems involving arc lengths, central
angles, and the radius in a circle.


radians are related to the radius
according to the diagram below, we can see the relationship between the radius, arc length, and 1 radian
An arc of length ____ of a circle of radius ____ subtends (or opens up on
to) an angle of ____ radian.
In other words, 1 radian is the measure of a central angle that is subtended
by an arc equal in length to the __________________________________.

How many degrees does it take to “go around a circle?”

How many times does the radius “go around a circle?”
HINT: Use the diagram below (the radius is r). Graphically,
the radius will wrap around the circle as follows.
Converting Radians to Degrees
Converting Degrees to Radians
If 2 radians = 360, then
 360  
180  
1 radian = 
=
  
 2 
If 360 = 2 radians, then
 2 
  
1 = 
radians = 
radians

 360 
180 
In general,
In general,
What is the size of 1 radian in degrees?
NOTE: Any angle measurement given without a unit is assumed to be in radians.
i.e.   2 means  = 2 radians .
e.g. Convert each of the following to radians or degrees. Leave all radian measurements in terms of  .
(a) 30º

(b) 2
7
(c) 6
(d) 225º
(e) 2 radians
(f) 252º
a


consider a circle with radius r, and an arc of length a that subtends a central angle 
the equation for calculating the arc length is a  r where the angle 
is measured in radians

r
e.g. Determine the arc length in a circle of radius 10 cm if:
a) the central angle is 5 radians
b) the central angle is 25º
e.g. Determine the central angle (in degrees) subtended by an arc of length 3 cm in a circle of radius 8cm .



co-terminal angles are standard position angles that share a common terminal arm
we are not limited to the number of times we can rotate the terminal arm of  , either clockwise or counter
clockwise
such rotations produce an infinite family of co-terminal angles


in general, consider an angle  in standard position measured in radians
co-terminal angles of  measured in radians will have the form  s  n2 where n is any integer

if  is in degrees, co-terminal angles of  will have the form  s  n360 where n is any integer
e.g. Determine the smallest positive and negative co-terminal angles for the following standard angles.
a) 127
c)

9
4
b)

3
d) 
3
2