Introduction To Radians
I. Angles As Ratios
The angle subtended by an arc at the center of a circle can be expressed as the ratio of the arc of a circle and it's radius. No matter how large or small the circle is, the ratio of the arc and the radius will remain constant as long as you keep the angle constant.
Try the following widget to get a better feel of this.
You will observe that when you change the radius of the circle, keeping $\angle POQ$ constant, the ratio of the arc $PQ$ and the radius $OP$ remains constant.
So, for any given arc $S$, radius $r$ and angle $\theta$, we can say that $S \propto r$ where the angle $\theta$ is constant, or $S \propto \theta$ when $r$ remains constant, and when both $r$ and $\theta$ are varying we can say:
$S \propto r \theta$
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II. Understanding Radians
In the previous section we arrived at the proportionality relation $S \propto r \theta$. Introducing the proportionality constant, we get the equation:
$S = k r \theta$.
Like we normally do, we can select our unit such that the proportionality constant gets eliminated. To do so, we can choose the unit of angle such that an angle of measure one unit is equal to the angle formed by an arc of length equal to the radius, that is $S = r$
Substituting $S = r$ and $\theta = 1$ in the above equation we get $k = 1$.
This unit is called the $\text{radian}$ and we get the relation $S = r \theta \Rightarrow \theta = \dfrac{S}{r}$ when $\theta$ is expressed in radians.
$\therefore Angle\ in\ radians = \dfrac{Length\ of\ Arc}{Length\ of\ Radius} = \dfrac{\overset{\frown}{RS}}{\overline{OR}}$
In the figure below, length of the arc $RS$ is equal to the radius $OR$. Remember that the length of the arc $RS$ is more than the straight line $RS$. Length of the line segment is denoted by $\overline{RS}$ and the length of the arc is denoted by $\overset{\frown} {RS}$
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By definition, $\angle ROS = 1\ radian$
III. Some Facts About Radians:
$\pi$ radians make a straight angle or $180^\circ$.
The number $\pi$ is an irrational number, whose approximate decimal value is $3.14159$ and in fraction, it is commonly taken as $\dfrac{22}{7}$. On some occasions it is taken as $\dfrac{355}{113}$. For all our problems we will take the value of $\pi$ as $\dfrac{22}{7}$, unless mentioned otherwise.
The common angles are denoted as follows:
$\dfrac{\pi}{2}\text{ radians} = 90^\circ$
$\dfrac{\pi}{3}\text{ radians} = 60^\circ$
$\dfrac{\pi}{4}\text{ radians} = 45^\circ$
$\dfrac{\pi}{6}\text{ radians} = 30^\circ$