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

2. Bellwork

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

4. 6.1 Radian and Degree Measure
Standard Position:
An angle is in standard position when its vertex is at the origin and its initial side lies on the positive x-axis.
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.

8. 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.

9. Radian and Degree Measure
Measuring Angles

10. 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

11. 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.

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

13. Find one positive angle and one negative angle that are coterminal with (a) −45° and (b) 395°.
There are many such angles, depending on what multiple of 360° is added or subtracted.

14. 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

15. 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.

16. Radian and Degree Measure
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r.
In general,

17. Radian and Degree Measure
Radian Measure

18. Radian and Degree Measure
Radian Measure

20. 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

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

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

23. Bellwork Convert the degrees to radian measure. 1. 30 -54
MAT 200
Bellwork
Convert the degrees to radian measure 
-54
Convert the radians to degrees. 3. 4.

25. 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.

26. 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

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

28. Bellwork Convert from degrees to radians. 54 -300
Convert from radians to degrees. 3. 4.

29. A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle θ of a sector is the angle formed by the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians.

30. A softball field forms a sector with the dimensions shown
A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field.

31. In Exercises 33–38, use a calculator to evaluate the trigonometric function.