y’’ = -y
Radians are God’s chosen angular unit. If you want to do mathematics, you have to use radians.
Radians are God’s chosen angular unit. If you want to do mathematics, you have to use radians.
There’s a log2 function and a log10 function and they are both useful. But when we talk about the log() function there can be no doubt that it is to base e.
If you want to define a sinT() function that works in turns then that’s totally fine. But the sin() function is defined as taking an argument scaled in radians, because it is mathematically natural.
Mathematically speaking, all trig functions are in radians. When you write sin(90°) the degree symbol ° is a conversion factor. I blame calculators for confusing high schoolers into believing that there is a separate set of functions that work in degrees.
[0] unsurprisingly because Euler’s formula equates the trigonometric and exponential functions.
The other sine function, defined using right triangles takes argument in Angles, also has nothing to do with the measuring unit.
(Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment, but definitely not universal between fields and languages. Personally I prefer ln, ld and lb for natural, decimal and binary logarithms.)
e.g. https://reference.wolfram.com/language/ref/Log.html
But I must confess that we had ln() in university courses and by default log used base 10. Now I use ln and a base for the log as a subscript like log_10, log_2, etc.
> But I must confess that we had ln() in university courses
Same. I often wonder why would anyone denote the natural logarithm with log(), when ln is shorter, and easier to read (at least for the people that were thought to use it), also it is already somewhat established.
The point is that a definition of the sine function where sin(pi/2) = 1 is equivalent to a sine function taking radians.
You could also define sinT(x) such that sinT(1/4) = 1: sinT(x) = sin(2pi x) = sin (tau x) = 2pi * x - (8pi^3 * x^3) / 3! + [...]. Neither of these is more or less fundamental than the other, but one is more convenient in most (non-trig) calculations.
This is a nearly universal convention in modern mathematics (except a few niches like information theory and computational complexity theory where it means log base 2, which is usually clear from context).
Engineering disciplines used to use "common" logarithms (i.e. base ten) all over the place back when most calculations were done with slide rules, lookups in paper tables, and pen-and-paper arithmetic, but with the advent of computers multiplication is just as cheap as addition, and expressing things on log scales is less necessary.
Over time the mathematicians are winning the fight to define the symbol 'log'.
https://mathworld.wolfram.com/Logarithm.html
https://mathworld.wolfram.com/CommonLogarithm.html
https://encyclopediaofmath.org/wiki/Logarithmic_function
https://encyclopediaofmath.org/wiki/Logarithm_of_a_number .
Also me, and the other commenter below learned "ln" for the natural logarithm, and used "log" only for binary/decimal logarithm.
I am starting to think that log = log_e is actually the minority usage, but it is certainly not universal.
In my opinion, mathematicians always choose the notation that's more convenient for them, at the moment, for a particular problem.
If a given problem is easier using another form of sin/cos, etc., they will use it, and it will be used without hesitation. In that sense, mathematicians could not be more pragmatic.
However, for many things, as long as the result is correct, they don't care about the operations' computability. Performance is an afterthought because for them (a*п)/п is exactly the same as 'a'. All operations are instantaneous.
Taylor series for example are a perfectly fine final answer in calculus, but to a programmer they are an infinite set of partial approximations that can take any arbitrary time to execute.
This is what makes computer science fascinating =)
have a `sin(x)` where the unit of x is radiants and a `sin_turn(x)` where x is expressed in turns.
Video games especially are a great situation to do it like this because they often use a framework (game engine) that was specifically created for this purpose.
I'm a software engineer, not a mathematician. Context matters.