Avoiding Trigonometry (2013)

If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.

An undecidable proposition is neither true nor false, it is not both true and false.

A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.

Thus what the previous poster has said has nothing to do with what Gödel had proved.

Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).

The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.

The trigonometric functions need not inject worse approximations than division.

If you compute trigonometric functions where the arguments are binary floating-point numbers and you measure the angles in cycles, not in radians (using radians is always a huge mistake in my opinion), the results can be expressed exactly using rational operations and the sqrt function.

You could compute them symbolically and use such symbolic expressions for exact computation, like you use rational numbers.

If you compute them numerically, computing a sqrt does not need more time than a division and correct rounding or computing an arbitrary number of digits are also not more difficult than for division.

Of course, you typically do not care about this, so you can just compute the trigonometric functions approximately, like you also do with division and sqrt, and in a similar time.

If you want consistency, you should measure all angles in cycles, not in radians.

Degrees are better than radians, but usually they lead to more complications than using consistently only cycles as the unit of measure for angles (i.e. to plenty of unnecessary multiplications or divisions, the only advantage of degrees of being able to express exactly the angle of 30 degrees and its multiples is not worth in comparison with the disadvantages).

The use of radians introduces additional rounding errors that can be great at each trigonometric function evaluation, and it also wastes time. When the angles are measured in cycles, the reduction of the input range for the function arguments is done exactly and very fast (by just taking the fractional part), unlike with the case when angles are measured in radians.

The use of radians is useful only for certain problems that are solved symbolically with pen on paper, because the use of radians removes the proportionality constant from the integration and derivation formulae for trigonometric function. However this is a mistake, because those formulae are applied seldom, while the use of radians does not eliminate the proportionality constant (2*Pi), but it moves the constant into each function evaluation, with much worse overhead.

Because of this, even in the 19th century, when the use of radians became widespread for symbolic computations, whenever they did numeric computations, not symbolic, the same authors used sexagesimal degrees, not radians.

The use of radians with digital computers has always been a mistake, caused by people who have been taught in school to use radians, because there they were doing mostly symbolic computations, not numeric, and they have passed this habit to computer programs, without ever questioning whether this is the appropriate method for numeric computations.

That's right. However, one disadvantage of using the tan value (the y/x ratio) over that of (cos,sin) tuple is that the former loses information on whether the y coordinates or the x coordinate was negative.

So, if you use the tan representation you have to carry that information separately. Furthermore, the code needs to correctly handle zero and infinity.

Tan of the half angle takes care of the first problem and is related to the stereographic transform. This works modulo one full rotation.

Depends on the hardware type.