Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
Radians are not a real unit. It’s more ratio of arc length to radius (m/m = dimensionless). That in itself has interesting properties, for example when converting angular speed in rad/s (so really /s) to linear speed at a wheel’s circumference (v = angular speed x radius)…among others.
Math works out better.
2 pi radians to define a circle works real well when you are using trigonometric functions and especially limits that arise from calculus.
An infinite Taylor series for the sin function looks much more elegant and easier to understand in radians than the one in degrees. The degrees one has a constant factor on every term that highly implies you should be using radians.
Thats the general gist of it. When mathematics bridges from the world of sinusoids to other things the equations are much less complicated and full of scaling factors if you use radians.
They exist because we defined them, I guess. I don’t really know how to answer your first question.
Radians much more convenient and natural for most areas of maths. Beyond simplifying length and area calculations on a circle, properties like the derivative of sin x being cos x, or the limit of sin x / x at 0 is 1 only work when x is expressed in radians. And when so much of maths relies on calculus, you’re pretty sure people will want to go with the choice that’s most elegant.
Of course nothing fully breaks if you insist on using, say, degrees here, but you end up with factors of pi/180 everywhere.
Because radians are a useful value defined by the radius of the circle, and degrees are an arbitrary value defined by how many days ancient people thought there were in a year.
In geometry and trigonometry, you use Pi a lot. You’re often doing math that concerns dealing with areas and arcs of circles, which are usually defined in terms of Pi. For example, the area of a circle is the radius squared times Pi. That means if you want the area of a part of the circle, you’re still going to be working with Pi.
If you express angles in degrees, the math needs to involve a lot of decimal places very quickly. But radians express angles in terms of Pi. So when you end up dividing something with Pi in it by something else that is a ratio of Pi, the math is a lot easier. Pi has a lot of decimal places, but saying “Pi / 2” is a lot easier than a number with four decimal places.
So in engineering you might still see people use degrees because that’s more convenient for architectural diagrams and other things that use degree-based tools like protractors. Those people have calculators and will do the math with a lot of decimal places.
But in math papers, when people are talking theories and proofs, they’ll use radians because they’re trying to effectively summarize a concept and it’s easier to visualize if the yucky decimal places get abstracted into a symbol. You probably have no clue what 29.6088132 / 6.28318531 is, but if I say “3 times Pi squared / two times pi” it makes more sense and the answer is more clearly simplified to “three times Pi over two” and that’s relatively easy to calculate as about “9.3 and some change / 2 = 4.65”.
Notice my 4.65 estimate’s a good bit off from the exact 4.77 of the first one. That’s what using fewer decimal places does and why engineers use calculators. Mathematicians don’t bother going a step further than “three pi over two”.
The radians are in many ways the “natural” unit for angles. This makes them very convenient especially in modelling the physical world. The arc length of one radian of a circle with a diameter of one feet is one feet (or one metre is one metre). This links angular and linear displacements, velocity and accelerations without annoying conversion factors.
The radians, however, are also a bit of a nuisance for mental math. Which is probably the primary reason we use multiple different units for angles.
A good example of such an intermediate unit is “mil”, which is approximately 0.001 radians. Ten mils at a kilometre equals ten metres.
As a sidenote, “mils” have been so convenient that they were invented several times independently. There are actually three different surviving definitions for “mils”. The most “accurate” splits the circle into 6300 mils (as 2000π ≈ 6283 and a spare milliradians). But, the two more popular ones choose to sacrifice some accuracy for some convenience. Some say 6000 mils to a circle, and others 6400 mils to a circle. The latter is more accurate, but still allows some easy subdivisions. The former is a bit less accurate, but is very well divisible just like 60 minutes are, or 360° are.
They’re more useful when doing calculations. For example, if you’re calculating the area of a slice of a circle you basically have to convert the slice from degrees to radians.
In the military we used milliradians because it increases proportionally so its easier to estimate distance/length based on length/distance. 1 mil = 1 meter at 1 kilometer. so if you know a tank is 10 meters long and you look through your M-22 binoculars and notice that it takes up 1 mil in your reticle you can estimate the distance at 10 kilometers.
The relationship of a circle’s radius r and arc length s is s = rθ where θ is the angle in radians. This relationship defines that 1 radian is the angle where the radius and arc length are equal. This makes the radian a very natural choice for angles. For example, since 2π radians is the angle subtended for a full circle, the arc length is just the circumference 2πr.
The mathematical series that can be used to calculate trigonometric functions work directly with radians. It’s slower to calculate these functions in degrees because you need to convert to radians first, requiring an extra multiplication.
Also, for small angles, θ≈sin(θ)≈tan(θ), but, of course, only if you work in radians. Radians is the natural way, with 2π units per revolution, as compared to 360 units which, while conveniently divisible by many numbers, is still just arbitrary.
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