Degree, Radians and their Conversions

Degree, Radians and their conversions

Angle is generally measured in degrees or radians. Degree is written using symbol (° ) and radians is written without any symbol. 360° make up one circle.

Radian: one radian is the measure of an angle subtended at the centre of circle by an arc length equal to radius of circle.

Using this we have relation as,

$\dpi{120} \mathbf{\theta =\frac{s(arc length)}{r(radius)}}$

Relation among three measures of angles:

360° = 2π radians = 1 revolution

Here is an useful table showing relation among the three measures of angle(revolutions, degrees and radians)

words	rev	deg	rad

no turn	0	0°	0
quarter turn	1/4	90°	π/2
half turn	1/2	180°	π
three-quarter turn	3/4	270°	3π/2
full turn	1	360°	2π

Revolutions(turns) are a more natural unit of measure than degrees. To convert revolutions to degrees we multiply with 360 and to convert degree to revolutions we divide by 360.

Radians and degree conversion :

We know that,

360° = 2π

1° = 2π/360 = π/180

To convert degree to radians we multiply with π/180 and to convert radians to degrees we multiply with 180/π .

DEGREES to RADIANS: RADIANS to DEGREES :

Degree*π/180 = Radians Radians*180/π = Degrees

Example: Example:

120° =120*(π/180) = 2π/3 3π/4 =(3π/4)*(180/π) =3*45=135°

Example: Convert the following degrees into radians.

Solution: a) 22.5°

$\dpi{120} 22.5^{\circ}=\frac{225}{10}*\frac{\pi }{180}=\frac{225\pi }{1800}=\frac{\pi }{8}$

b) 315°

$\dpi{120} 315^{\circ}=315 *\frac{\pi }{180}=\frac{315\pi }{180}=\frac{7\pi }{4}$

Example : Convert the following radians into degrees.

Solution: a) 2π/15

$\dpi{120} \frac{2\pi }{15}=\frac{2\pi }{15}*\frac{180}{\pi }=\frac{360}{15}=24^{\circ}$

b) 7π/8

$\dpi{120} \frac{7\pi }{8}=\frac{7\pi }{8}*\frac{180}{\pi }=\frac{1260}{8}=157.5^{\circ}$

An angle in degrees can be written completely using minutes and seconds. As

1 = 60minutes (‘)

1’ = 60 seconds (”)

Lets work on some examples on how to convert radians into degrees, minutes and seconds.

Example: Convert the following radian measures into degree, minutes and seconds.

Solution:

a) π/8

$\dpi{120} \frac{\pi }{8}*\frac{180}{\pi }=\frac{45}{2}= \left (22\frac{1}{2} \right )^{\circ}=22^{\circ}\left ( \frac{1}{2}*60 \right )'=22^{\circ}30'$

b) 1/4

$\dpi{120} \frac{1}{4}*\frac{180}{\pi }=\frac{45}{\pi }=\left ( 45*\frac{7}{22} \right )=\left (\frac{315}{22} \right )^{\circ}=\left ( 14\frac{7}{22} \right )^{\circ}=14^{\circ}\left ( \frac{7}{22}*60 \right )'$

$\dpi{120} = 14^{\circ}\left ( 19\frac{1}{11} \right )'=14^{\circ}19'\left ( \frac{1}{11}*60 \right )''= 14^{\circ}19'\left ( \frac{60}{11} \right )''=14^{\circ}19'5''$

c) -2

$\dpi{120} -2 *\frac{180}{\pi }=\left ( \frac{-2*7*180}{22} \right )^{\circ}=\left ( -114\frac{6}{11} \right )^{\circ}=-114^{\circ}\left ( \frac{6}{11}*60 \right )'$

$\dpi{120} =-114^{\circ}\left ( 32\frac{8}{11} \right )'=-114^{\circ}32'\left ( \frac{8}{11}*60 \right )''=-114^{\circ}32'44''$

Convert minutes and seconds into degrees

$\dpi{120} 1'=\left ( \frac{1}{60} \right )^{\circ}$

$\dpi{120} 1''= \left ( \frac{1}{60} \right )'= \left ( \frac{1}{3600} \right )^{\circ}$

Lets work on some examples on how to convert minutes and seconds into degrees . Here is a video example.

Example : Convert the following degree, minutes and seconds into decimal degrees.

Solution:

1) 25°12′

$\dpi{120} 25^{\circ}12'= 25^{\circ}+\left ( \frac{12}{60} \right )^{\circ}=25^{\circ}+0.2^{\circ}=25.2^{\circ}$

2) 42°15’45”

$\dpi{120} 42^{\circ}15'45''=42^{\circ} +\left (15* \frac{1}{60} \right )^{\circ}+\left ( 45*\frac{1}{3600} \right )^{\circ}$

= 42° + 0.25° + 0.0125° = 42.2625°

Practice problems:

Convert the given angle measures from radians into degrees

18π/5
11

Convert the given angle measures from degrees to radians

–56°
47.5°

Convert the given angle measures from radians to degrees,

Minutes and seconds

6
π/32

Convert the given angle measures from degree, minutes and seconds to degrees only

40°20‘
5°37‘30”

Answers:

648°
630°
–14π/45
19π/72
343°38‘11”
5°37‘30”
(121/3)°
(45/8)°