Do You Find Converting Degrees to Radians Confusing?

I did too. But recently, I learned about it and it’s pretty simple. Let us understand systematically how we can convert degrees to radians and radians to degrees. First, it is important for us to know what radians and degrees are. The definition of an angle tells us that a unit is one complete revolution from the position of the initial side. There are two units of measurement of an angle that is most commonly used. They are radian measure and degree measure. Let us now discuss these units of measurement in detail.

Converting Degrees to Radians

Radian measure = π/180 × degree measure.

Let us see an example: Convert 40°20’ into a radian measure.

We know that 180  =  radian

Hence 40°20’= 40 1/3 degree = /180 × 121/3 radian = 121 π/540 radian

Therefore, 40°20’=121π/540 radian.

How to Convert Radians to Degrees?

Let us understand what radians and degrees are and then we will proceed on to how to convert radians to degrees.

• Radian Measure: Radians are one of the standard mathematical measures to calculate the measurement of an angle. The angle formed at the centre by an arc of length one unit in a unit circle is said to have a measure of one radian. We know very well that the circumference of a circle of radius one unit is 2π. Therefore, one complete revolution of the initial side forms an angle of 2π radian. It is very well known to us that equal arcs of a circle form an equal angle at the centre. Now, since in a circle of radius r, an arc of length r forms an angle whose measure is 1 radian, an arc of length l will form an angle whose measure is l/r radian.

Thus, we get l = r × θ.

• Degree Measure: It is the most commonly used measure to calculate the measurement of an angle. The angle is said to have a measure of one degree if a revolution from the initial side to the terminal side is 1/360^th of a revolution. A degree is divided in 60 minutes and a minute is divided into 60 seconds.

Converting Radians to Degrees

Let’s see how to convert radians to degrees

Degree measure = 180/π × Radian measure

Let us see an example: Convert 6 radians to degree measures.

We know that π radian = 180°

Hence, 6 radians = 180/π × 6 degree =1080×7/22 degree

= 343 7/11 degree= 343 + 7 × 60/11 minute

=343 °+38’+2/11 minute

=343° + 38’ +10.9” =343°38’11” approx.

Hence, 6 radians =343°38’11” approximately

Relationship Between Degree and Radian

Since a circle forms an angle at the centre whose degree measure is 360° and whose radian measure is 2π radian, we can say that

2π radian = 360° or π radian = 180°

Therefore, 1 radian = 180°/π

Also 1° = π/180 radian.

Some Common Angles Used in Trigonometry

30 degrees = π/6 radian

45 degrees = π/4 radian

60 degrees = π/3 radian

90 degrees = π/2 radian

180 degrees = π

360 degrees = 2π

Notational Convention

Since angles are either measured using degree measure or the radian measure, there is a convention in place of writing them all over the world. Whenever you write angle θ°,it means the angle whose degree measure is θ and whenever you write angle β, it means the angle whose radian measure is β.

History of Origin of Degrees and Radians

The selection of the number of degrees as 360 is believed to have its roots in the methods of the ancient Babylonians. As evidenced in the Rig Veda, the division of a circle into 360 equal parts also occurred in Ancient India. The concept of using the length of an arc for the measurement of an angle was first made popular in the early 1700s.

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