[Maths Class Notes] on Degrees to Radians Pdf for Exam

Degrees and radians are two different units that are used for the measurement of the angles. The conversion of degrees to radians is considered while measuring the angles in Geometry. The measure of the angle is generally denoted by degrees, having the symbol °. An angle can be determined by two different kinds of units, that are, degrees and radians. You can convert one form of the representation of any mathematical angle to the other by using simple formulas. A degree also has its sub-parts that are minutes and seconds. This conversion plays a major part in the trigonometry applications. In this article, we will learn about how to convert degrees to radians, degrees to radians formula, and look at some solved examples based on how to convert degrees to radians formula. Let us first look at the degree to radian conversion.

The Degree to Radian Conversion

The value of 180° equals [pi] radians. For converting any given angle from the measure of its degrees to the radians, you need to multiply the value by [frac{pi}{180}].

The value of [pi] is [frac{22}{7}] or 3.14. 

Degrees to Radians Formula

We already learned about how to change degrees to radians for any given angle. Let us learn about how to convert degrees to radians formula. The formula to change the degree to radian is given as follows:

Degree [times  frac{pi}{180}]  = radians

How to convert Degrees to Radians

Let us now look at a step by step procedure for converting degrees to radians.

1. Jot down the degrees that you want to convert into radians. Consider the following examples: 

Example 1: 120°

Example 2: 30°

Example 3: 225°

2. Then multiply the degrees by [frac{pi}{180}]. 

Example 1: [120 times frac{pi}{180}]

Example 2: [30 times frac{pi}{180}]

Example 3: [225 times frac{pi}{180}]

3. Then simply carry out the multiplication by multiplying the degrees by π/180. Think of it as if you are multiplying two fractions. The first fraction consists of the degrees in the numerator and 1 as the denominator, and the second fraction consists of π in the numerator and has 180 in the denominator.

Example 1: 

[120 times frac{pi}{180}]

= [frac{120pi}{180}]

Example 2: 

[30 times frac{pi}{180}]

=[frac{30pi}{180}]

Example 3: 

[225 times frac{pi}{180}]

= [frac{225pi}{180}]

4. The last step is to simplify. Now, you have to put each fraction in its lowest terms to get the final answer. Find the largest number which can evenly divide to the numerator and the denominator of each fraction and use it for simplifying each fraction. 

Example 1: 

[120 times frac{pi}{180}] 

=  [frac{120pi}{180} ÷ frac{60}{60} ] 

= [ frac{2}{3 pi} ] radians

Example 2: 

[30 times frac{pi}{180}]

= [frac{30pi}{180} ÷ frac{30}{30} ]

= [frac{1}{6pi}] radians

Example 3: 

[225 times frac{pi}{180}]

= [frac{225pi}{180} ÷ frac{45}{45}  ]

= [frac{5}{4π}] radians

This is a very simple method and you can easily convert degrees to radians with this simple procedure. Let us look at how to convert angles to radians in detail.

How to Convert Angles to Radians

You learned about how to change degrees to radians. Now let us learn about how to convert the angle to radians. 

The angle that is made when the radius is wrapped around the circle is given as follows:

1 radian is approximately 57.2958°.

Consider a half-circle which has radians, which equals to 180°.

Hence, 1 radian = [ frac{180^{0}}{pi} ] = 57.2958°

If you want to convert a degree or an angle to radians, simply multiply the angle by and then divide it by 180.

Take a look at the table below of the angles and their conversion to radians.

To convert  degree measure into radian measure, students may directly use the formula. Multiply the given value in degrees with [frac{pi}{180}]. This is a simple step and students can use this to find radian measure. However, the table given above tells the radian and approximate radian values of the most common angles. Students may use this table for easier and faster calculations. For example, if a student wants to calculate the radian measure of 30°, 60° and 90°, he or she may refer to the table. The radian measures of the following measures in degree will be [frac{pi}{6}], [frac{pi}{3}] and [frac{pi}{2}] and the radian values will be 0.524, 1.047 and 1.571 respectively. 

Uses of Radian

1. Radian is another unit to measure angles and also the SI unit of measurement of angles. It is defined as the angle made at the center of the circle by an arc which is equal to the length of the radius of th
   e circle. 

2. It is denoted by ‘rad’ or by using the letter c. Angle written without a unit means it is written in radian measure. Some examples are 4 rad, [frac{pi}{2}]  or 90° 

3. Calculus and other fields of mathematics use radians as the unit of measurement. It is also used in the areas of science. 

Solved Examples

Let us now look at some of the solved examples for your better understanding of how to convert degrees to radians and radians to degrees.

Example 1

Convert 120° to radians.

Solution:

To convert 120° to radians, consider the formula

[text{Angle in radians} = text{angle in degrees} times  (frac{pi}{180})]

Therefore, [120° times (frac{pi}{180}) ]

= [(frac{2pi}{3})] radians ≈ 2.09 radians

Example 2

Convert 1.4 radians to degrees.

Solution:

To convert radians to degrees, consider the following formula:

[text{Degree} =  text{radians}  times  frac{180}{pi} ]

= [1.4 times 180 = 252 ]

= [1.4 times frac{180}{pi} = frac{252}{pi} ]

≈  80.2° 

Example 3

Convert [frac{4pi}{9} ] radians to degrees. 

Solution:

To convert radians to degrees, students may use the following formula:

[text{Degree} =  text{radians}  times  frac{180}{pi} ]

= [frac{4pi}{180} times frac {180}{pi} ]

= [frac{4pi}{180} times frac {180}{pi} ]

= [4pi times frac {180}{pi}  times pi]

= [4 times 20]

= 80°