Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
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”.
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