Last time, What’s the deal with 360?, we asked the question “Why are there 360 degrees in a full circle?”. The answer we got was that there isn’t really one - 360 is a bit arbitrary, kind of plucked out of nowhere. So is there a better way to measure angles?
This is where radians come into the story. First things first radians are simply another way to measure angles; we have meters or feet to measure distance, Celsius or Fahrenheit to measure temperature etc, and now degrees or radians to measure angles.
So how do we define a radian? Well here goes: one radian is defined to be the angle inside a sector whose arc length is precisely equal to the radius of the circle. Phew. Confused? What this means is that we take a circle and measure the radius with a piece of string. We then wrap this string part way around the outside of the circle - the angle we create is what is defined to be one radian! Watch the animation below to see this visually, the red line is our piece of string…
This animation also shows us something else; there are 2pi radians in a full circle. This is because the circumference of a circle is given by 2pi times the radius (of course). This gives us the conversions rates: 2pi radians = 360 degrees, 180 = pi, 90 = pi/2 and even the other way round; 1 radian = 180/pi degrees, or approximately 57.296 degrees.
So that’s defined a radian but now what’s the big deal about it? Why is it any better than using degrees?
The thing to notice is that in our definition of a radian we only used things already given to us (a radius and a circle) unlike with degrees where we pluck the number 360 out of nowhere. This makes radians the “natural” way to talk about angles. Think about our alien that’s visiting us again, we struggled to answer his question about why we use 360 degrees but we could easily tell him why we use radians.
When we start to do mathematics with this it becomes even more apparent why radians are the most natural way to talk about angles, things just really work - and boy do they work beautifully.
Let’s look at just one single example. Here’s the Taylor Series for Sin, when we’re measuring ‘x’ in radians (here comes an equation but don’t panic!)…
The Taylor Series for Sin, with the angle x measured in radians.
It doesn’t even matter if you don’t get what this equation means or where it comes from, just look what a beautiful and simple equation we have. But this simplicity is because we’re measuring our angle, x, in radians. If we try to do the same with degrees things fall apart a little…
And again, but for the angle x measured in degrees this time.
Look how much uglier this formula has gotten - it’s awful! I mean heck, it doesn’t even fit on one line. We’re having to introduce these strange constant factors on each term to force things to work. Even if you don’t really understand these (and it doesn’t matter if you don’t, these are university level formulas!) which one would you rather use? Which one looks nicer? Which one seems natural? And so from this which way of measuring angles is more natural?
This really was just one example, there’s countless other formula whose beauty simply falls apart if you try to use degrees!
So the take home lesson, the moral of the story, for today is that radians are defined naturally and because of this doing mathematics with them just works.
As a final note, let’s be honest degrees are here to stay and so they should; they’re completely ingrained in our culture and history and they’re so simple and easy to understand, I mean try teaching radians to a 8 year old and you’re going to struggle! But once we want to do any serious mathematics using anything but radians is pure crazy!