1. Radian and Degree Measure
MAT 200
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
Reference angles

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

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

5. 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. Radian and Degree Measure
Measuring Angles

7. 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. 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. 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.
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. 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. Radian and Degree Measure
Radian Measure

14. Radian and Degree Measure
Radian Measure

15. 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. MAT 200
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. Reference Angles

21. Reference Angles
The values of the trigonometric functions of angles greater than 90  (or less than 0 ) can be determined from their values at corresponding acute angles called reference angles.

22. Reference Angles
Figure 4.37 shows the reference angles for  in Quadrants II, III, and IV.
Figure 4.37

23. Example 4 – Finding Reference Angles
Find the reference angle  .
a.  = 300  b.  = c.  = –135 
Solution:
a. Because 300  lies in Quadrant IV, the angle it makes with the x-axis is
  = 360  – 300 
= 60 .
Degrees

24. Example 4 – Solution
cont’d
b. Because 2.3 lies between  /2  and   , it follows that it is in Quadrant II and its reference angle is
  =  – 2.3 
c. First, determine that –135  is coterminal with 225 , which lies in Quadrant III. So, the reference angle is
  = 225  – 180 
= 45 .
Radians
Degrees

25. Example – Finding Reference Angles
Find the reference angle  ′. a.  = 300 b.  = 2.3 c.  = –135

26. Example (a) – Solution
Because 300 lies in Quadrant IV, the angle it makes with the x-axis is
 ′ = 360 – 300
= 60.
The figure shows the angle  = 300
and its reference angle  ′ = 60.
Degrees

27. Example (b) – Solution
cont’d
Because 2.3 lies between  /2  and   , it follows that it is in Quadrant II and its reference angle is  ′ =  – 2.3  The figure shows the angle  = 2.3 and its reference angle  ′ =  – 2.3.
Radians

28. Example (c) – Solution
cont’d
First, determine that –135 is coterminal with 225, which lies in Quadrant III. So, the reference angle is  ′ = 225 – 180 = 45. The figure shows the angle  = –135 and its reference angle  ′ = 45.
Degrees

29. Your Turn: Find the reference angle for each of the following. 213°
1.7 rad
−144°
-144 ̊ is coterminal to 216 ̊
216 ̊ ̊ = 36 ̊

30. Example 4 – Solution
cont’d
Figure 4.38 shows each angle  and its reference angle  .
(a)
(b)
(c)
Figure 4.38

31. Reference Angles Review
Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ
ref angle
ref angle
ref angle
ref angle
Think of the reference angle as a “distance”—how close you are to the closest x-axis.

32. Reference Angles
When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2𝜋.

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

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

35. Video Link www.youtube.com/watch?v=wcfkDuFpbiM