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