Radian and Angles

September 1, 2020

Reading Time: 5 minutes

Introduction

Radian, as a concept, was first described by Roger Cotes back in 1714. But the term 'Radian' first appeared in print in 1873. The name radian was not universally adopted for a while, in fact, Longmans' school of trigonometry still called radian the 'circular measure' when published in 1890.

 
           

 As shown in the picture,

• There are 2 pi radians in a circle (2 * 3.14159..). That equates to 6 complete radians and a little more than a quarter of a radian (0.2831853..). 
• The length of each arc (s) subtended by an angle of one radian is equal to the radius itself. s = r ( for 1 radian arc).
• The magnitude of such a subtended angle (in radians) is equal to the ratio of the arc length subtended to the radius of the circle; which means θ=s/r, where θ is the subtended angle in radians.
• The length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radian; that is s=rθ.

Also read:

• Trigonometric Identities

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What is a Radian? 

Radian is the plane angle subtended by a circular arc and its value is given by the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
The unit is also called a circular measure of an angle.

Unit symbol: The international bureau of weights and measures specify rad as a symbol of the radian. Alternative symbols used 100 years ago are c (the letter), or a  superscript R.

Radian
Unit System	SI derived unit
Unit of	Angle
Radian Symbol	rad  or c
In unit	dimension with the arc length equal to the radius i.e 1mm

 

Conversions
Units	1 rad is equal to
Milliradians	1000 mrad
Turns	12π turns
180 degrees	180 =57.296˚
gradians	200=63.662g

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Unit Circle

The unit circle is a circle with the value of its radius as 1 unit. It makes learning and talking about lengths and angles much simpler.
The x- and y-axes divide the entire coordinate plane into four quarters (90˚ each) called quadrants. The trigonometry quadrants are as follows:

Quadrant I	0-90˚ or (0 - π/2) rad
Quadrant II	90-180˚ or (π/2 - π) rad
Quadrant III	180-270˚ or (π - 3π/2) rad
Quadrant IV	270-360˚ or (3π/2 - 2π) rad

Any angle with a value greater than 360˚ or 2π can be calculated by subtracting a multiple of 360, which is also the closest value lesser than the angle itself from it so that we can reduce all angle values to a range of (0,360) or (0,2π).

Eg:  17.5π = 17.5π - 8*2π = 1.5π or 270˚

For any angle, we can label the side and the unit circle by its coordinates(x,y). The coordinates x and y will be the output of the trigonometric functions

where,
X=cos t; Y=sin t

The diagram of the unit circle  is as follows: 

         

  
Coordinates of a point on a unit circle where the central angle is t radians: The value of x and y is given by the length of the two triangles legs. This is a right triangle, and it displays how the lengths of these two sides are given by trigonometric functions of t.

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Degree to Radian Conversion

1 rad = 180/π
1 rad= 1 * (180˚)/ π  ≈  57.2958˚ 

Angle in degrees = Angle in radians * (180˚)/π

For example,

1. 2.5 rad =  2.5 * (180˚) /π 
                ≈ 143.2394˚
2. π/3 rad = π/3* (180˚)/π
                 ≈ 60˚

To convert from degrees to radians, multiply the angle in degrees by π/180 such that,

Angle in radians= angles in degrees * π/(180˚)

For example:

1. 1˚ = 1 * π/(180˚) 
        ≈ 0.0175 rad
2. 23˚= 23 * π/(180˚) 
         ≈ 0.4014 rad

Radian is converted to turns by dividing the number of radians by 2π.

Radian to degree conversion derivation

The circumference of a circle is 2πr               (r is the radius of the circle)

Using this,

360˚↔ 2πr                                                      (since a complete circle sweeps 360˚)

By the definition of radians, a full circle has:
2πr/r  rad= 2π rad
Combining both the above relations

2π rad=360˚
≡ 1 rad = (360°)/2π
≡ 1 rad = (180°)/π
Example 1:  Convert 50˚to radians
Answer:   50˚ x π/180 = 50π/180 = 5π/18

Example 2:  Convert π/6 to degrees
Answer:      π/6 x (180˚)/π = 180/6 = 30˚

If an angle is given as a value without the degree symbol, then the angle is generally assumed to be in radian measure.

Conversion of common angles:

 

Turns	Radians	Degrees
0	0	0˚
1/24	π/12	15˚
1/12	π/6	30˚
1/10	π/5	36˚
1/8	π/4	45˚
1/2π	1	57.3˚
1/6	π/3	60˚
1/5	2π/5	72˚
1/4	π/2	90˚
1/3	2π/3	120˚
2/5	4π/5	144˚
1/2	π	180˚
3/4	3π/2	270˚
1	2π	360˚

 

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What is the need for Radian?

Since radian is the angle subtended by an arc equal in length to the radius of the circle, the definition is far less arbitrary than a degree so you could claim it is a better unit purely. 
Both radian and degree are units of angle measurement, but the angle 360 in degree is arbitrary since there is no mathematical base to choose that value from.
Also, 1 radian = 2*pi…where pi comes in a picture. Pi is the dearest number to mathematics and we are talking about circles, who else stands a chance?

 

• A one-radian arc has an arc length equal to the radius
• It also makes the value of trigonometric functions of very small angles much easier to assume and compute.
• It makes differentiation of trigonometric functions more straight forward.
• It makes radial motion formulas, and some harmonic motion formulas, more straightforward as well.  

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Summary

A radian equal to 57.3˚ and an angle subtended by an arc whose length is equal to the radius of the circle.
Radian = length of arc/ radius of circle 
Unit circle is a circle with a radius of 1. It is a great way to learn and talk about lengths and angles
The length of the circumference of a circle is given by 2πr, where r is the radius of the circle The radian is  the angle of a circle subtended by an arc  equal in length to the radius
 one radian arc has an arc length equal to the radius. 

Written by Sakshi Goel, Cuemath Teacher