Rational values of trigonometric functions

Niven's Theorem: If $x/\pi$ (in radians) and $\sin x$ are both rational, then the sine takes values $0$, $\pm 1/2$, and $\pm 1$.

Obviously, angle in radians is a rational multiple of $\pi$ iff angle in degrees is rational.

Let us take the Pythagorean theorem $a^2 + b^2 = c^2$ and divide both sides by $c^2$ to get $\frac{a^2}{c^2} + \frac{b^2}{c^2} = 1$. Now, we can use the well-known relationship between sine and cosine: $\sin^2(x) + \cos^2 (x) = 1$, and let $\sin(x) = \frac{a^2}{c^2}$, and $\cos(x) = \frac{b^2}{c^2}$.

Since the hypotenuse of a triangle is always longer than the two legs, $\frac{a^2}{c^2}, \frac{b^2}{c^2} < 1$. Therefore, there exists a bijection between one triplet of $a,b,c$ and $\frac{a^2}{c^2}, \frac{b^2}{c^2}$, and at least one value of $\frac{a^2}{c^2}, \frac{b^2}{c^2}$ for $\sin(x)$.

One example using a $3,4,5$ triangle shows that $\cos(x) = \frac{9}{25}$, and using the second equation $\sin(x) = \frac{16}{25}$. As a bonus, since there are infinitely many Pythagorean triples, this shows that there are infinitely many values of $\sin(x)$ and $\cos(x)$ that are rational.