1. Radian and Degree Measure
MAT 200
Radian and Degree Measure
In this section, we will study the following topics:
Vocabulary used to describe angles
Degree measure of an angle
Radian measure of an angle
Converting between radian and degree measure
Find coterminal angles

2. Radian and Degree Measure
MAT 200
Radian and Degree Measure
Important Vocabulary for this Lesson:
Angle
Degrees
Radians
Angle in Standard Position
Coterminal angle
Reference Angle
Initial Arm
Terminal Arm

3. Radian and Degree Measure
Angles
Trigonometry: measurement of triangles
Angle Measure

4. Radian and Degree Measure
Standard Position
Reference Angle
Vertex at origin
The initial side of an angle in standard position is always located on the positive x-axis.

5. Radian and Degree Measure
Positive and negative angles
When sketching angles, always use an arrow to show direction.

6. Radian and Degree Measure
Measuring Angles
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees and in radians.
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of of one revolution.

7. Radian and Degree Measure
Measuring Angles

8. Radian and Degree Measure
Classifying Angles
Angles are often classified according to the quadrant in which their terminal sides lie.
Ex1: Name the quadrant in which each angle lies.
50º
208º II I
-75º III IV
Quadrant 1
Quadrant 3
Quadrant 4

9. Radian and Degree Measure
Coterminal Angles
Angles that have the same initial and terminal sides are coterminal.
Angles  and  are coterminal.

10. Radian and Degree Measure
Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º.
In general, to find coterminal angles,
θ = ± 360ºn
where n is a positive integer
Ex 2: Find one positive and one negative angle that are coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

11. Ex 3. Find one positive and one negative angle that is coterminal with the angle  = 30° in standard position.
Ex 4. Find one positive and one negative angle that is coterminal with the angle  = 272 in standard position.

12. STOP! Time to do an activity!

13. Radian and Degree Measure
Radian Measure
In the activity you did, the angle AOB that you found has a measure of 1 radian
Definition of Radian:
One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.
In general,

14. Radian and Degree Measure
Radian Measure

15. Radian and Degree Measure
Radian Measure

16. Radian and Degree Measure
Conversions Between Degrees and Radians
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by

17. Ex 5. Convert the degrees to radian measure.
60
30
-54
-118
45

18. Ex 6. Convert the radians to degrees.
a) b) c) d)

19. Ex 7. Find one positive and one negative angle that is coterminal with the angle  = in standard position. Hint: for coterminal angles in radians, θ = ± 2πn
Ex 8. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

20. Degree and Radian Form of “Special” Angles
0° 
360 ° 
30 ° 
45 ° 
60 ° 
330 ° 
315 ° 
300 ° 
 120 °
 135 °
 150 °
 240 °
 225 °
 210 °
 180 °
90 ° 
270 °  
Degree and Radian Form of “Special” Angles

21. Class Work Convert from degrees to radians. 54 -300
Convert from radians to degrees. 3. 4.

22. Find one postive angle and one negative angle in standard position that are coterminal with the given angle.
135