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Indeed, maths never "decreed that sine and cosine have to take radian arguments". But thinking that makes any sort of point is a fundamental misunderstanding of
by pdpi 4y ago
Indeed, maths never "decreed that sine and cosine have to take radian arguments". But thinking that makes any sort of point is a fundamental misunderstanding of maths.
There are infinitely many sinusoidal functions out there. You can just adjust amplitude, frequency and phase to your heart's content.
Trigonometry basically requires that sine and cosine have specific amplitudes and phases, but gives not one shit about how you map angles to frequency. Degrees are completely arbitrary, but both radians and turns have pretty natural definitions, with turns indeed being the easiest to work with. So far so good.
Calculus does have an opinion on frequency, though. There is exactly one non-trivial pair of sinusoids s(x) and c(x) where c'(x) = - s(x) and s'(x) = c(x), among a bunch of other very useful properties.
When you put calculus and geometry together, s and c are have the same amplitude and phase as sine and cosine from geometry, and the two pairs are exactly the same if you match the frequencies such that the argument is the angle measured in radians. It's just so damned useful to use angles in radians and make everything play together nicely.
- deleted 4y ago[deleted]
- jacobolus 4y ago> Degrees are completely arbitrary Degrees are very natural in the context of ancient astronomy/astrology, where you have (1) ~365 days in a year, so that if you look at the path of something that takes a year you get about one degree change per day but with a number that is more easily divisible. (2) approximately 4y, 10y, 8y, 15y, 12y, 30y cycles for the moon and various planets. (3) A calendar with 12 months, 12 zodiac signs. (4) A timekeeping system which breaks days into 24 hours and then uses divisions by sixty for smaller units. (4) A base-sixty number system – from ancient Mesopotamia, which persisted as the standard for astronomical calculations for millennia, only displaced in the very recent past.
- sicariusnoctis 4y agoAll those approximations have error and the behaviour you describe depends on where you are on Earth. Moreover, degrees are not natural from a mathematical perspective.
- lupire 4y agoThat's... convenient for approximately our specific arbitrary position in the universe, but not mathematicallly natural.