1. 6.1 Radian and Degree Measure
MAT 200
6.1 Radian and Degree Measure
In this section, we will study the following topics:
Terminology used to describe angles
Degree measure of an angle
Radian measure of an angle
Converting between radian and degree measure
Find coterminal angles

2. 6.1 Radian and Degree Measure
Angles
Trigonometry: measurement of triangles
Angle Measure

3. 6.1 Radian and Degree Measure
Standard Position
Vertex at origin
The initial side of an angle in standard position is always located on the positive x-axis.

4. 6.1 Radian and Degree Measure
Positive and negative angles
When sketching angles, always use an arrow to show direction.

5. 6.1 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.

6. 6.1 Radian and Degree Measure
Measuring Angles

7. 6.1 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

8. 6.1 Radian and Degree Measure
Classifying Angles
Standard position angles that have their terminal side on one of the axes are called quadrantal angles.
For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.

9. 6.1 Radian and Degree Measure
Coterminal Angles
Angles that have the same initial and terminal sides are coterminal.
Angles  and  are coterminal.

10. 6.1 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.
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. 6.1 Radian and Degree Measure
Radian Measure
A second way to measure angles is in radians.
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,

13. 6.1 Radian and Degree Measure
Radian Measure

14. 6.1 Radian and Degree Measure
Radian Measure

15. 6.1 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

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

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

18. Ex 7. Find one positive and one negative angle that is coterminal with the angle  = in standard position.
Ex 8. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

19. 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

20. MAT 200

21. Reference angles
the angles formed between the terminal side of an angle in standard position and the closest side of the x-axis.
All reference angles measure between 0o and 90o
You will use reference angles when filling out the unit circle.
They are helpful because you can memorize less of the UC if you use reference angles.

22. Quadrant I
Quadrant II
Quadrant III
Quadrant IV
45o
120o
275o
195o

23. Reference Angles To find a reference angle in…Q1
Do nothing!
45o Q2
subtract the angle from 180o
180o - 120o = 60o Q3
subtract 180o from the angle
195o - 180o = 15o Q4
subtract the angle from 360o
360o o = 85o

24. Practice: Find the reference angles
870o
-45o
-200o
-900o
-314o
300o
210o
135o
585o
30o

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

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

27. HW p odd, 37-41odd, 43-47odd