Radians and DegreesIntro to Calculus

March 2010

The trigonometric functions are among the most fundamental in mathematics. They were initially developed to aid in the measurement of triangles and their angles, and they are used daily in surveying and navigation. However they can be used to describe any natural phenomenon that is periodic, and in higher mathematics they are fundamental tools for understanding many abstract spaces.

Wicked Important!

The primary use of trigonometric functions is in the measurement of angles. Although the ancient Babylonian degree unit of angle measure is still in wide use, in mathematics we prefer to use the radian measure. Given a circle centered at the origin in the Cartesian plane, imagine taking a radius and laying it along the outside circle, beginning at the x axis and going counterclockwise.

We begin with two basic measurements followed by two fundamental definitions.

Once around a circle is 360º.

And that the circumference of a circle is 2r

Some basics…

If the radius is 1, C =2(1)

Circumference or 360º = 2

Using the fact that 360º corresponds to 2Π radians, we can generate the following angle measures:

Dividing equals 180º which is radians

Dividing gives us 90º which is radians

Degrees to Radians

2

2360º

2180º

Using the fact that 360º is 2 radians and 180º is radians, we can always convert degrees to radians by

Multiplying

the degree measurement by

Degrees to Radians

º

Converting degrees to Radians!

If we have a radian measure we can multiply by

So

And we can convert to degrees …

180

60

160180180

Degree measure of angle is based upon the in a circle and radian measure is based upon as another way to describe one complete circle.

We can convert from radians to degrees

And from degrees to radians.

Wrap Up

rad

34º

4π rad

46º

rad

rad

3π rad

π rad

150º

rad

732º

6π rad

rad

2010º

Convert to radians or degrees!

23

32

4

5

Convert to radians

52º

34º

35º

4π rad

74º

36º

15º

37º

94º

53º

174º

156º

376º

324º

163º

532º

272º

631º

856º

428º

732º

994º

897º

1768º

2000º

Convert to degrees

2π rad

π rad

3π rad

4π rad

7π rad

9 rad

4π rad

π rad

14π rad

8π rad

3π rad

25π rad

45π rad

6π rad

73π rad

8 rad

2π rad

6π rad

5π rad

3π rad

The unit circle and Radians