Day 1 - Angles & Angle Measures
Degrees
Radians
–
–
What you are used to
A new way to measure angles
What is a Radian?
The angle AOB that you created in the investigation measures 1 radian.
A radian is related to the length of the arc of a circle and the radius of that circle.
One full rotation: 2  R  360
One half rotation:  R  180
One quarter rotation:
One eighth rotation:
Important Angles to Know:
Degree
Radian
Angles Extended:
Degree
Radian
180°
120°
360°
150°
90°
210°
270°
300°
60°
135°
30°
225°
45°
315°
Is there a conversion factor that will help convert degrees to radians?
What if I asked you the exact radian measure of 138°=
75°=
290°
To change from radians to degrees:
4R 
2.7 R 
3

7
6

5
(Note: we don’t have to put the radian symbol when using radians)
Conversion Factors (unit conversion)
Radians to degrees
Degrees to radians
Arc Length
As a result of the definition of a radian, a relationship between the arc length of a circle
and the angle subtended by the radii can be determined:
arc length
angle at centre

circumference
2
a


2r 2
a  r
Example 1
A circle has a radius of 5.5 cm. Calculate the length of an arc of this circle subtended by
each angle. Express length to the nearest centimetre.
a) 3.5 radians
b) 215
Draw an angle of 30°.
In geometry – the orientation of the angle doesn’t matter – in trigonometry, we position
the angle on a coordinate plane and call it an angle in standard position.
Draw 30° in standard position.
(Important to include point P and direction of rotation)
0
Terminology:
Initial arm
Terminal arm
Quadrants
Angles that have the same initial and same terminal arm are called coterminal
angles.
On the diagram above draw and give the measure of three coterminal angles to 30°.
Give a general way of stating all angles which are coterminal to 30°.
Example 2
Sketch the angle –110° in standard position. Draw and give the measure of two
coterminal angles. Give a general solution for all coterminal angles to –110°.
0
Example 3

in standard position. Draw and give the measure of two coterminal
3

angles. Give a general solution for all coterminal angles to .
3
Sketch the angle
0
Example 4

in standard position. Draw and give the measure of two coterminal
4

angles. Give a general solution for all coterminal angles to  .
4
Sketch the angle 
0
Example 5
A point P is on the terminal arm of an angle  in standard position that has been rotated
15

about the origin.
4
a) How many rotations have been made?
b) In what quadrant is P located?
c) Draw a diagram to show the position of P
Summary:
An angle  between the initial arm and terminal arm, which represents the amount of
rotation about the origin, is called the angle in standard position. The measure of the
angle in standard position can be in radians or degrees.
If   0 , the rotation is counterclockwise.
If   0 , the rotation is clockwise.
This motion is repeated every 360 0 or 2 ; therefore, is periodic.
Coterminal Angles
When a position P is described in more than one angle of standard position the angles are
said to be coterminal angles.
Coterminal angles can be represented by
  n(360 0 ) , in degrees, where n is any integer ( n I )
  2n , in degree, where n is any integer ( n I )