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 $90^\circ$?
Best Answer
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.