Btukfyl

What are the possible combinations of sine, cosine, and tangent values such that all three are simultaneously rational?

I am aware of the below two cases.

• $sin(x) = 0, cos(x) = 1, tan(x) = 0$ is a trivial case for angle measure zero




• $sin(x) = 0.6, cos(x) = 0.8, tan(x) = 0.75$ is another case, for an angle of $37$ degrees (approximately)

Are there any other combinations that satisfy the condition? If not, is it possible to prove that these are the only possible combinations?

Any Pythagorean triple will give you rational $sin, cos, tan$ values, and any triple of rational trig values will give you a Pythagorean triple.

The Pythagorean triples have been completely classified. Take two natural numbers $u>v$. Then
$$
(u^2-v^2)^2 + (2uv)^2=(u^2+v^2)^2
$$
and any Pythagorean triple is either of this form or a multiple of such a triple. So the possible rational $sin$ and $cos$ pairs are all possible values of
$$
frac{u^2-v^2}{u^2+v^2}text{ and }frac{2uv}{u^2+v^2}
$$

A hypotenuse-$1$ right-angled triangle containing an angle $x$ has side length $sin x$ opposite it and $cos x$ adjacent to it, and if these are rational so is their ratio $tan x$. But these lengths are rational iff the triangle can be scaled to have integer side lengths, so the rational choices for $sin x,,cos x,,tan x$ with $x$ acute are precisely those obtained from Pythagorean triples. Extending to arbitrary angles so some functions may be negative, the general solution is $sin x=frac{2ab}{a^2+b^2},,cos x=frac{a^2-b^2}{a^2+b^2},,tan x=frac{2ab}{a^2-b^2}$ with $a,,binmathbb{Z},,anepm b$.

Note that

$$frac{sin(x)}{cos(x)} = tan(x)$$

Therefore, whenever sine and cosine are rational, and cosine nonzero, tangent is also rational. Since sine and cosine are also continuous for all real numbers $x$ and are what we call "onto" or "surjective" on the interval $[-1,1]$, for each number in that interval we can find $x$ such that $sin(x)$ and $cos(x)$ are that number.

We also know that the rationals are "dense" in the reals - that is, in between any two real numbers, you can always find a rational number. It ergo goes that we can find infinitely many $x$ combos so that $sin(x)$ and $cos(x)$ (independent of each other) are rational.

This doesn't really address the case of their ratio or them being rational at the same time, though.

For this, as noted by Blue in the comments, Pythagorean triples will always work. Pythagorean triples define three sides of a right triangle, specifically positive integer solutions to $a^2 + b^2 = c^2$. Because the triangle is a right triangle, we can also note that any $x$ we use as the angle here (opposite a leg) will be between $0$ and $pi/2$ in measure, noninclusive.

There is a method to generate infinitely many Pythagorean triples too! Take positive, unequal integers $m,n$ - any two you want. Then $a = 2mn, b = m^2 - n^2, c = m^2 + n^2$. Since $m,n$ are integers, then $a,b,c$ are integers. Then in turn $sin(x) = b/c$ (or $a/c$, depending on the choice of angle), and $cos(x) = a/c$ (or $b/c$), giving those two are rational. From that, we see that $tan(x) = b/a$ (or $a/b$) and thus is also ratio.

Thus, for each pair of unequal, positive integers $m,n$ we can have

$$sin(x) = frac{b}{c} = frac{m^2 - n^2}{m^2 + n^2} ;;; cos(x) = frac{2mn}{m^2 + n^2} ;;; tan(x) = frac{m^2 - n^2}{2mn}$$

giving infinitely many solutions such that $sin(x),cos(x),tan(x)$ are all rational. From there, finding the corresponding $x$ is trivial.

As for whether these are the only solutions, I do not know.