Have you ever wondered why we have different units to measure certain things? For example, temperature is measured in Fahrenheit and Centigrade, Capacity is measured in liters and cubic meters, distance in miles and kilometers etc. We shall add one more to this list - the angles - commonly measured in degrees and radians. Why do we need two different units to measure the same quantity?
Most of the quantities that were listed above have geographical impacts. However, using radians in measuring angles have special reasons.
First let us understand the meaning of degrees and radians. Watch my video here.
We can see that degree is an arbitrary measure of an angle, ie., we could have cut the circle in many other sections, not necessarily 360. However, why we chose 360 is for another discussion!
Now let's see what a radian is. Radian is formed by a specific sector - in which the arc length is equal to the radius of the sector. With this theory, we see that the number of sectors of angle one radian is an irrational number = 2π!!! I think this is the most interesting part of this concept where you actually talk about number of sectors being irrational. This makes radians act more like a real number on a number line! That's probably why we are able to use them on the x-axis when we plot trigonometric functions.
We start using radian measure once we start learning trigonometry. We can say that the coordinates of a point on a x-y plane can be expressed as (cosθ, sin θ), where θ is the angle that the position vector of the point makes with the positive x-axis. And we start drawing the graphs of the sine and cosine curves as the point moves along a circle in the anti-clockwise direction. This is when we shift to radians. Though both degrees and radians can be used in this context, using radians takes us to the next level of trigonometry, viz, trigonometric functions.
Trigonometric ratios become functions only when we start using the angles as radians. For example, Sin(x) is considered as a 'continuous' function which is 'differntiable' only when x is expressed in radian. The reason being, the limit of sin(x)/x as x tends to 0 is equal to 1 only when x is in radians.
Radians are also helpful in Euler's formula eiθ = Cos θ +i Sin θ, that can be easily interpreted using radian measure for the angle. The famous result eiπ= Cos π +i Sin π, can not be expressed in degrees. If you look into the proof using Taylor's expansion for this result, the expansion is possible only when the x in ex is a real number.
There are more interesting areas where the radians are more helpful than degrees while dealing angles.
