Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Degree & Radians

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

A degree is divided in 60 minutes and  a minute is divided in 60 seconds

 1^o=60',\quad 1'=60"

Some angles are shown below (In degree)

Radian Measure

Angle subtended at the centre by an arc of length  1 \text{ unit} in a unit cirlce (circle of radius  1  \text{ unit}  is said to be a measure of   1 radian.

The circumference of a circle of radius  1  \text{ unit} \text{ is }  2\pi . Thus, one complete revolution of the initial sides subtends an angle of   2\pi radian.

In a circle of radius  r , an arc of length or subtends an angle whose measure is  1 radian, an arc of length   l will subtend an angle whose measure is  \dfrac{l}{r}\text{ radian} .

Thus, is a circle of radius  r  , an arc length length  l subtends an angle  \theta  radian at centre, we have

 \theta =\dfrac{l}{r}\text{  or } l=r\theta

Some angles whose measures are in radian shown below:

Relation Between Degree and Radians

 2\pi \text { radian }= 360^o

Or  \pi\text { radian}= 180^o \left( \therefore \pi=\dfrac{22}{7}\text { or } 3.1415\right)

  \implies 1 \text { radian}= \dfrac{180^o}{\pi}=57^o16'(\text{ approx.)}

Also,  1^o=\dfrac{\pi }{180}\text { radian }= 0.01746 \text{ radian} (\text{ approx.)}

Degree  30^o  45^o 60^o  90^o 180^o  270^o  360^o
Radian  \dfrac {\pi}{6}  \dfrac {\pi}{4}  \dfrac {\pi}{3}  \dfrac {\pi}{2}  \pi  \dfrac {3\pi}{2}  2 \pi

Notational Convention

  \text{Radian Measure}= \dfrac{\pi}{180} \times \text{ Degree Measure}

 \text{Degree Measure}= \dfrac{180}{\pi} \times \text{ Radian Measure}

Example:

Convert  40^o into radian.

We know  180^o=\pi \text{ radian}

 \implies 40^o=\dfrac{\pi}{180} \times 40 =\dfrac{2\pi}9 \text{ radian}

Example:

Find the radius of the circle in which a central angle of  60^o  intercepts an arc of length  37.4\text {c.m.}\left( \pi=\dfrac{22}{7} \right)

Solution:

Here,  l=37.4 \text {c.m.}, \quad\theta =60^o=\dfrac{\lambda}{3} \text{ radian}

Hence,   r=\dfrac{l}{\theta}

 =\dfrac{37.4}{\dfrac{\pi}{3}}=\dfrac{37.4 \times3}{\dfrac{22}{7}}=35.7\text { c.m}.

\therefore r=35.7 \text{ c.m.}

Example:

Convert  \dfrac{3\pi}{8}   radian in degree measure

Solution:

We know  \pi  \text{ radian}= 180^o

 \begin{aligned} \implies \dfrac{3\pi}{8}\text{ radian} &=\dfrac{180^o}{\pi}\times\dfrac{3\pi}{8}\\&=22.5^o\times 3=67.5^o \\&=67^o30' \end{aligned}

\therefore  \dfrac{3\pi}{8}\text{ radian =67}^o30'

Scroll to Top