Rational values of trigonometric functions

Question

I am using extensively trigonometric functions when an angle is given in degrees.
Some of these functions like sine or cosine have rational values, for example, the well known example is that $\cos(\theta) =0.6 $ and $\sin(\theta) =0.8 $.

However besides the case of multiplicity of $90^\circ$ it seems there are no rational numbers $\theta$ with simultaneously rational values of sine and cosine.

• Is it possible somehow to prove that for rational values of an angle given in degrees there are no values simultaneously rational of sine and cosine functions, beside obvious case of multiplicities of $90^\circ$?

Answer 1

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.

Answer 2

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.