Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
Very simply: degrees are an arbitrary unit. We could decide on 300 or 42 degrees to a circle. There is nothing specific about 360 that would lead a separate group to use that number of degrees.
But there is a reason that another group would use 6.28 radians to describe a full circle. This is the relationship between the traits and diameter. This is what you get no matter how you measure the circle.
It’s kind of the same reason we have fractions and decimals. They are both useful in different scenarios. If you want to have a ration way to represent 1/3, we have the fraction to do that. If you want to have a value for an angle that doesn’t have pi attached to it at every level, we have degrees. But decimals and radians are much easier to work with in many systems of math.
Radians are basically the native angle measurement inherent in a circle that enable direct translation of angles to other aspects of geometry, and through this are the native angle measurement inherent in trigonometry where the math presents much more elegantly than using degrees, especially when you start to get into complex numbers and calculus.
For a basic example:
How long is the arc of a circle radius R drawn by an angle “A” – in Radians this is simply R times A. Similarly using radians the area of the sector enclosed by this angle is simply R^2 times A/2. If you measure the angle in degrees you have to do other gymnastics to get to the same place.
The simplest reason they’re more useful: if I want to know the length of an arc spanning an angle x, then the length is just x*R if x is in radians – which lines up (by definition) with the total circumference of the circle being 2πR. Doing the same thing in degrees involves needing to also multiply by π/180 and just makes things look messy. This tidiness of the relationship between angle in radians and arc length shows up all over the place so often radians are more convenient for doing math, even if degrees are nicer to read off of a protractor sometimes.
Pi (π) is the ratio of a circle’s circumference (the length around its border) and its diameter (the length of any line drawn from one point on the circle border to the directly opposite point. This line splits the circle into two perfectly even halves).
Written as an equation where c is the circumference and d is the diameter,
c = π × d
Because circles often the whole or entirety of something, and because circles show up frequently in the math relating to cycles, many equations have pi in them. Defining the angles of a circle using pi therefore makes sense, since it keeps the equations simple/straightforward to solve.
When you ask your scientific calculator co compute, for example, cosine of 48.5-degrees, the calculator logic actually converts to radians first before generating the polynomial sequence which will compute the answer. Radians are the natural unit.
Cosine 48.5-degrees = Cosine 0.8464-radians = 1 – (08464)^2 /2! + (08464)^4 /4! – (08464)^6 /6! + (08464)^8 /8! – etc.
If you stand on the corner of a big pie wedge, and measure the angle in radians, you’ll immediately know the length of the crust is the angle x radius. you don’t have the additional steps of multiplying by pi and dividing by 360.
If you cut the crust off to make the pie wedge a triangle, you can’t do this for the straight edge normally. However, if the angle is small enough, it’s good enough for government work. So if you stand at the top of a narrow triangle with two equal legs, the opposite side is pretty much the angle x leg length if the angle is less than 0.5 radians or so.
Now you’ve got a great way for quickly figuring out how big something is at a distance. If you take a piece of glass at arms length and scratch out angles in radians, you can figure out how tall something is if you know they’re 1000 feet away. Just multiply the angle x 1000 feet and you’ll get their height. You could do the reverse, if you know their heigt, divide it by the angle and you get the distance, and you never had to muck about with pi or 360. Note you can do this with any distance unit, eg feet, meters, inches, whatever, you’ll get your answer in the same units.
If we know that the ratio of the diameter (2R) and circumference (I) of a circle is a constant and call it pi = I/2R then 2Rpi = I. If we introduce the radius R as a characteristics distance for a circle we can talk about the ratio of I and R. 2pi = I/R. If we don’t fix I to be the circumference but allow some portion of it p we get 2pi p = Ip/R. The number on the left side works how we’d want an angle to work. The total distance covered on the circumference pI = 2pi p R.
Very convenient isn’t it this number which ranges from 0 to 2pi describes a full rotation around a circle and can be turned into distance travelled around the circumference by multiplying with the radius of the specific circle we are looking at. So the length of a circular curve is just the internal angle that traces the curve times the radius. You won’t get a more natural way of introducing some measure of angles than this.
sin(x) = x where x is small.
This is a very, very useful property.
Also helps with you are dealing with calculus – calculus functions with points ends up in giant messes.
I think this gif illustrates it very well
https://upload.wikimedia.org/wikipedia/commons/4/4e/Circle_radians.gif
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