Today I was wondering the following thing (I won’t bore you with how I ended up with this question):
Are there any rational values of for which the line is tangent to the graph of
Clearly the answer is yes: But my gut feeling was that this was the only such After some head scratching, I obtained the following proof.
Suppose they are tangent at . Then the point of tangency is Moreover, the slope of the tangent line is . Thus, must satisfy the two equations:
Using a trig identity,
Note that is an algebraic number—it is the root of a polynomial with integer coefficients.
In 1882 Ferdinand von Lindemann famously proved that is transcendental (that is, non-algebraic) for every non-zero algebraic number and a similar proof holds for the sine and cosine functions.
Thus, because and is algebraic, it must be the case that . In particular,
This concludes the proof.