# 7. Radians

## 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***.*

**Care with your calculator!** Make sure your calculator is set
to **radians** when you are making radian calculations.

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

## Converting Degrees to Radians

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

2

πradians `= 360^@`.

This gives us the important result:

πradians = 180^{o}

From this we can convert:

radians → degrees and

degrees → radians.

### Example 1

Convert the following to degrees:

a. `1` radian

b. `2` radians

Answer

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

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

Please support IntMath!

### Example 2

Convert the following to radians:

a. `50^@`

b. `357^@`

Answer

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

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

Get the Daily Math Tweet!

IntMath on Twitter

### 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.

Answer

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"`

**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^@`

Answer

**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`

Answer

** 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^@`

Answer

`168.7^"o"=(168.7xxpi)/180=2.944`

Notice there is no degree sign after the answer. It is an angle in **radians**.

Easy to understand math videos:

MathTutorDVD.com

4. Express in terms of degrees: `1.703` radians.

Answer

`1.703=(1.703xx180)/pi=97.57^"o"`

Get the Daily Math Tweet!

IntMath on Twitter

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Trigonometry Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand trigonometry lessons on DVD. See samples before you commit.

More info: Trigonometry videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!