Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
Radians are really useful when you need to convert from a rotating context to a straight line (linear) context, and vice versa.
For example, if you have a wheel with a radius of 1 meter, this means that each time the wheel rotates 1 radian, it also travels forward 1 meter. So if the wheel rotates at 1,000 radians per hour, we immediately know that we are also moving forward at 1,000 meters per hour; also known as 1 kilometer per hour.
This context switching between rotating and linear contexts happens *a lot* in engineering, so it’s really convenient to have a unit of measure that expresses angles in a way that we can very easily convert to linear measurement.s
If you want to know how far a wheel has traveled, it’s the angle of rotation (in radians) times its radius. Every problem involving “rotating without slipping” needs angles in radians.
Also, if you learn a little calculus you’ll run across something called a Taylor’s expansion, which is a way of expressing trigonometric functions as polynomials; e.g., sin(x) = x – x^3 /3! + x^5 /5! -… These functions only work when x is in radians.
To your point, when performing trig functions of angles, it doesn’t really matter if you sin(45°) or sin(pi/4), as long as your calculator is in the right mode. But as soon as you start multiplying angles with other angles or most other math with them, the angles need to be in radians.
I think of it as angles in degrees have units, and angles in radians do not.
For small angles, say less than 5°, the angle in radians is equal to the sine. So there’s no need to convert to degrees. You just take a ratio and you’re done.
Let’s say something is 100 m away and has a height of 5 m. That means it occupies 5/100, or 0.05 radians (50 milliradians, or 50 mrad) of your field of view. To convert to degrees you have to divide by pi and multiply by 180, because 180° is equal to pi radians. That makes the angle 2.86°.
Why go through the extra step of converting to degrees when using units of radians is just as valid? For quick reference, I’ve memorized that 57° is 1 radian, and 1° is ~17 mrad. Because I’m fun like that.
The weird thing about both radians and degrees is that they are both extremely useful in their appropriate applications. As a species, we so rarely do this well, but the usefulness of radians to any associated math or science, and degrees to ease of communication and divisibility is the rare time we got them both kind of right. Think about this for a second, we could have had gradians instead of degrees. Can you imagine what kind of a nightmare that would have been? 400 gradians in a circle. Divisible by so few numbers compared to 360.
I’ve been on both sides of this. When I was a pilot full-time, degrees were extremely natural and easy to use. Now, as a software engineer that does a lot of lat/long and heading/course manipulation, I use radians and degrees almost interchangeably. Degrees are good for course and heading selection. When those values have to be mathed into vectors for calculations, trig functions are called, which almost always take radians. Still, I always prefer to think in degrees. Some aspects of flying never leave you.
Some people will tell you radians are not a unit. Do not listen to them. They are a unit of measurement. Dimensionless, sure, but still a unit. I can measure turn speed in radians per second, definitely a unit of measurement. Somehow “I was turning at 0.2, uh, <blank> per second” doesn’t work.
One more nugget:
In some dark corners of industry, there is such a thing as a Pi-Radian. Take a normal radians value and divide by PI. Circles are 2 Pi-Radians. Not 2 Pi, just 2. It makes some math easier. Now, I’m not saying anyone should use Pi-Radians, only that they would still be better than Gradians.
Rads.are fantastic for a ton of applied math… they are kind of a pain to learn but then they make everything really nice and tidy…
Totally worth it (they are super rad :p )
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