## Radians - an Alternative Measure for Angle

In science and engineering, radians are much more convenient (and common) than degrees. A radian is defined as the angle between 2 radii (radiuses) of a circle where the arc between them has length of one radius.

Another way of putting it is: "a radian is the angle subtended by an arc of length r (the radius)".

In the diagram, we can see the 2 radii of the circle and the subtended arc length of r.

One radian is about 57.3^@.

Radians are especially useful in calculus where we want to interchange angles and other quantities (e.g. length). For example, see how radians are required in Fourier Series. That stuff won't work if we try to use degrees.

Most computer programs use radians as the default.

Also, see this simple introduction to radians with an interactive graph.

Because the circumference of a circle is given by C = 2πr and one revolution of a circle is 360^@, it follows that:

2π radians = 360^@.

This gives us the important result:

From this we can convert:

### Example 1

Convert the following to degrees:

a. 1 radian

b. 2 radians

a. 1\ "radian"=(180^"o")/pi=57.29578^"o"

b. 2\ "radians"=(2xx180^"o")/pi=114.59156^"o"

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### Example 2

a. 50^@

b. 357^@

a. 50^"o"=50xxpi/180=0.8726646

b. 357^"o"=357xxpi/180=6.2308256

### Example 3

Convert the following:

a. 60^@ to radians

b. 3.1 radians to degrees

c. π/4 radians to degrees

d. 156.34^@ to radians

Your calculator can do these for you. However, you are encouraged to know what is happening under the hood.

1) 60^"o"=60xxpi/180=pi/3"rad"

2) 3.1xx180/pi=177.62^"o"

3) pi/4 xx 180/pi = 45^"o"

4) 156.34xxpi/180=2.7286\ "rad"

Easy to understand math videos:
MathTutorDVD.com

NOTE: We'll see some examples of trigonometric ratios of angles involving radians in the section Radians and the Trigonometric Ratios.

## Exercises:

1. Express in radian measure in terms of π:

a) 12^@

b) 225^@

a) 12^"o"=(12pi)/180=pi/15

b) 225^"o"=(225pi)/180=(5pi)/4

Easy to understand math videos:
MathTutorDVD.com

2. Express the following angles in degrees:

a) (7pi)/15

b) (4pi)/3

a) (7pi)/15=(7xx180^"o")/15=84^"o"

b) (4pi)/3=(4xx180^"o")/3=240^"o"

3. Express in radian measure (use decimals): 168.7^@

168.7^"o"=(168.7xxpi)/180=2.944
4. Express in terms of degrees: 1.703 radians.
1.703=(1.703xx180)/pi=97.57^"o"