If one wants to describe how far the edges of an angle are spread apart, the easiest concept to use would be degrees. As one progresses further into mathematics, one would quickly discover that there is in fact a totally different and confusing new unit for angles: the radian.
Nevertheless, without clear understanding of radians, trigonometry becomes even harder than it already is.
First and foremost, it’s important to clarify that the ultimate goal of radian is like that of degree: to measure angles. However, as its name suggests, the concept of radian comes from a different place: the radius of a circle. Before we get to that, we must discuss another prerequisite of radians: arcs of a circle.
Whenever we use degrees, we imagine the centre of a circle. Then, we extend two lines the given degrees apart from that centre. Thus, degrees look at the circle like a pie, and one can naturally cut away a certain amount from the centre. On the other hand, radians look at the circle like a string tied into a loop with empty space in the middle. Now, if one measures an angle, one must focus on the length of string taken away. This is the concept of arc length: how much length of a circle’s circumference.
All that radians do is to divide a circle’s entire arc length into units, and the length of each unit will be precisely the radius of the circle. You may have already reacted that the constant π multiplied by 2 is exactly how many radii equal to the circumference of a circle, and a circle should have 2π radians in total! This is exactly equal to 360° that normally makes up a circle.
Soon after one encounters radians, one would probably immediately start learning to convert between radians and degrees, but what is the point of radians? It turns out this approach of viewing an angle as an arc of a circle rather than the distance between two edges has surprising benefits.
The most basic advantage of radians is notational efficiency. In higher level mathematics, angles beyond 360° are common, but imagine having to count how many 360s are there in 4680°! In terms of radians, this would simply be 26π, and if you know 2π radians is 1 circle, you can easily deduce the number 13. Additionally, within a single 360°, it’s much easier to understand how far the angle goes in the circle with 5/4 π than 225°, given enough practice.
However, the most significant contributions of radians still remain in higher mathematical concepts such as trigonometric functions, complex numbers, and calculus in non-cartesian coordinate systems. These won’t be explained today. At least, you could finally evade the looming sense of purposelessness that I so deeply felt when studying radians in trigonometry for the first time.
Cover Image: "Love & Mathematics" by Lost Archetype is licensed under CC BY-NC-ND 2.0
“Positive Angle” by Gustavb is licenced under CC BY-SA 3.0
“Arc to Circle” by Nerd1a4i is licensed under CC BY-SA 4.0
“Radian” by Patrol 110 is licenced under CC BY-SA 3.0
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