Question

How to find the radian measure of any angle?

Answer 1

Hint: Consider the required angle as any variable. Apply unitary method to the basic relation between π and degrees. Once you get the radian relation, you can put any angle in place of the variable to find the radian measure of the desired angle.

Complete answer:
We know that the circumference of a circle is $2\pi r$,where r= radius of the circle.
Now, suppose we draw a circle with the radius of 1 unit (A unit can be anything, 1 meter, 1 centimeter or 1 kilometer) and measure an angle of 180° from the circle (it will be a semicircle), the length of the arc will be measuring π units.
Thus, we know that π radians =180 degrees.
Coming back to the question, let us consider any angle that is in degrees that we want to convert to radians as x.
Using unitary method, we can determine the x in radians as follows,
$
  \pi = 180^\circ \Rightarrow 180^\circ = \pi \\
   \Rightarrow 1^\circ = \dfrac{\pi }{{180^\circ }}radians \\
   \Rightarrow x^\circ = \dfrac{\pi }{{180^\circ }} \times x\,radians \\
 $
Thus, we see that for any angle x in degrees, ${\dfrac{\pi }{180^\circ } \times x}$ is the converted value in radians.

Note: The ratio of the length of a circular arc (a) to the radius of the arc (r) is known as the radian measure of an angle. Since, the ratio is defined from length (r) to length (a), radian measure is dimensionless and is a pure number. It is the SI unit of angular measure and is denoted as “rad”.