converting degree to radian, converting radian to degree

We already know how to measure an angle in degrees and radians. Let's revisit some of the concepts again.

Let the ray starts at original position OA and starts rotating. Its final position will be OB. The measure of an angle is the amount of rotation performed to get the terminal side(OB) from the initial side(OA). There are several units to measure the angle. The above figure shows one complete revolution from the position of the initial side(OA). If a rotation from the initial side to the terminal side is  \(\frac{1}{360}\) of a revolution, the angle is said to have a measure of one degree, written as 1^°. A degree is divided into 60 minutes and a minute into 60 seconds. 1 minute is written as 1' and one second as 1''.
1° = 60' and 1' = 60''

The below figures show angles whose measures as 360°, 180°, 90°, -30°.
Note: An angle is said to be positive if the direction of rotation is anticlockwise and negative if clockwise.

There is another unit for the measurement of angle, called the radian measure. The angle subtended at the Centre by an arc of length 1 unit in a circle of radius 1 unit is said to have a measure of 1 radian. The below figure shows angles of 1 radian and -1 radian.

O is centre of the circle, when OA is original position and move in anticlockwise to OB position. Length of arc AB = AC = 1 unit. Radius \(OA = OB = OC = 1 \) unit then measure of \(\angle BOA = \angle AOC = 1 \space \textrm {radian}\). We know that circumference of a circle of radius 1 unit is \(2\pi\). Thus one complete revolution subtends an angle of \(2\pi\) radian. If in a circle of radius r, an arc of length l subtends an angle \(\theta\) radian at the centre then
\(\theta = \frac{l}{r}\)
Since a circle subtends at the centre an angle whose measure is \(2\pi\) radian and its degree measure is 360°, therefore 
\(\mathbf{2\pi \textrm{ radian} = 360^\circ}\)

or

\(\mathbf{\pi \textrm { radian} = 180^\circ}\)

Assigning the value of \(\pi = \frac{22}{7}\), 1 radian = 57°16' (approx) and 1° =0.01746 radian (approx)

The relation between radian and the degree of common angles are given in the below table
 

Degree	30°	45°	60°	90°	180°	270°	360°
Radian	\(\frac{\pi}{6}\)	\(\frac{\pi}{4}\)	\(\frac{\pi}{3}\)	\(\frac{\pi}{2}\)	\(\pi\)	\(\frac{3\pi}{2}\)	\(2\pi\)

From the above table, we can derive the relation as:

Radian Measure \(\mathbf{ = \frac{\pi}{180}} \) × Degree Measure

Degree Measure \(\mathbf{ = \frac{180}{\pi} }\) × Radian Measure

Example 1: Convert 40° into radian measure.
Radian Measure = \(\frac{\pi}{180} \times 40 \) = \(\frac{2}{9} \pi\)

Example 2: Convert 6 radians into degrees.
Degree Measure = \(\frac{180}{\pi} \times 6 = \frac{1080 \times 7}{22} \)

=\(343\frac{7}{11} ^\circ\)
Break degrees into minutes and minutes into seconds

= 343 + ( 7 × 60) ∕ 11 = 343° + 38' + 2 ∕ 11''

= 343° + 38' + 11''

 Hence 6 radians = 343°38'11''(approx)