Angles

Fig-1 Positive Angle

Fig-2 Negative Angle

An angle is formed when two rays are joined together at a common point. The common point is called as node or vertex. An angle is a measure of rotation of a given ray about its vertex. Original ray is called as initial side and the final position of the ray after rotation is called as terminal side of the angle. The point of rotation is called as vertex. Angle is said to be positive if the direction of rotation is anticlockwise (Fig-1), angle is said to be negative if the direction of rotation is clockwise.(Fig-2). The measure of an angle is the amount of rotation done to get the terminal side from the initial side.

Degree Measure

If the rotation from initial side to terminal side is $\left({\frac {1}{360}}\right)$^th of a revolution, the angle is said to have a measure of one degree written as 1°.

$1\ degree(1^{\circ })=60\ minutes(60')$ $1\ minute(1')=60\ seconds(60'')$

Radian Measure

Radian is another unit of measurement of an angle. 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.

In a circle of radius $r$, an arc of length $l$ subtends an angle $\theta$ radian at the centre then

$\theta ={\frac {l}{r}}$ or $l=r\theta$

Fig.3-Radian

Relation between degree and radian

$Circumference\ of\ a\ circle=2\pi r$

if the $radius\ of\ a\ circle=1$ then $Circumference\ of\ a\ circle=2\pi$ . Hence one complete revolution of initial side subtends an angle of $2\pi$ radian and its degree measurement is $360^{\circ }$.

$2\pi \ radian=360^{\circ }$

$\pi \ radian=180^{\circ }$ , $\pi ={\frac {22}{7}}$

$1\ radian={\frac {180^{\circ }}{\pi }}={\frac {180^{\circ }\times 7}{22}}=57^{\circ }16'$ approximately

$\pi \ radian=180^{\circ }$ Hence $1^{\circ }={\frac {\pi }{180}}radian={\frac {22}{7\times 180}}=0.01746\ radian$ approximately.

The following table shows the relation between degree measures and radian measures of some common angles.

$Degree$	$30^{\circ }$	$45^{\circ }$	$60^{\circ }$	$90^{\circ }$	$180^{\circ }$	$270^{\circ }$	$360^{\circ }$
$Radian$	${\frac {\pi }{6}}$	${\frac {\pi }{4}}$	${\frac {\pi }{3}}$	${\frac {\pi }{2}}$	$\pi$	${\frac {3\pi }{2}}$	$2\pi$

$Radian\ Measure={\frac {\pi }{180}}\times Degree\ Measure$

$Degree\ Measure={\frac {180}{\pi }}\times Radian\ Measure$

Example 1:

Convert $45^{\circ }20'$ into radian measure.

$Radian\ Measure={\frac {\pi }{180}}\times Degree\ Measure$

$45^{\circ }20'=45{\frac {1}{3}}degree={\frac {136}{3}}radian={\frac {\pi }{180}}\times {\frac {136}{3}}={\frac {34\pi }{135}}\ radian$

Hence $45^{\circ }20'={\frac {34\pi }{135}}\ radian$

Example 2:

Convert $4$ radians into degree measure.

$Degree\ Measure={\frac {180}{\pi }}\times Radian\ Measure$

$={\frac {180}{\pi }}\times 4\ degree={\frac {180}{22}}\times 7\times 4={\frac {2520}{11}}=229{\frac {1}{11}}\ degree$

$1\ degree(1^{\circ })=60\ minutes(60')$

$=229\ degree+{\frac {1\times 60}{11}}\ minutes$

$=229\ degree+5{\frac {5}{11}}\ minutes$

$1\ minute(1')=60\ seconds(60'')$

$=229\ degree+5\ minutes+{\frac {5\times 60}{11}}\ seconds$

$=229\ degree+5\ minutes+27.2\ seconds$

Hence $4\ radians=229^{\circ }5'27''$ approximately.