# The representation of trigonometric functions of irrational multiples of $\pi$ in closed form

It is know that, if $\alpha\pi$ is an irrational multiple of $\pi$, where $\alpha$ is an algebraic number that can be expressed by radicals, then the trigonometric functions of $\alpha\pi$ cannot be expressed by means of radicals, since these values are transcendental numbers. However, consider the following formula

$$\cos \left ( \frac{\alpha}{n} \right ) =\frac{1}{2} \sqrt[n]{\cos \alpha + i \sin \alpha}+\frac{1}{2} \sqrt[n]{\cos \alpha - i \sin \alpha}$$

(A proper choice of the $n$th roots of unity is needed in order to get the right equality).

Now consider the number $\cos (30/\sqrt{2})^\circ$. If we use the formula given above, for $\alpha = 30^\circ$ and $n = \sqrt{2}$, we get

$$\cos \left ( \frac{30}{\sqrt{2}} \right )^\circ = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt{3}}{2}+\frac{i}{2}}+\frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt{3}}{2}-\frac{i}{2}} \tag 1$$

Evaluating both expressions, we get that $\cos (30/\sqrt{2})^\circ \approx 0.932240\ldots$ The other expression also has the same value.

This leads to the following questions. First, I have never seen a book that deals with the problem of the representation of $\cos (\alpha\pi)$ in closed form, where $\alpha$ is an irrational number that can be expressed by radicals. Trigonometric functions of these numbers cannot be expressed by means of radicals, but the situations seems to change if we allow to take $n$th roots to complex numbers, where $n$ is not a natural number, but any real number, as in the above example.

In fact, trigonometric functions of $\alpha\pi$, for any real irrational algebraic $\alpha$, seems to can be expressed in the form $\frac{1}{2} \gamma^\beta +\frac{1}{2} \gamma^{-\beta }$, where $\gamma$ is a complex number and $\beta$ is a real irrational algebraic number.

As these numbers are raised to an irrational exponent, they have infinitely many values. In fact, the left side of $(1)$ is single-valued, but the right side is multiple-valued: because of the multivalued nature of the natural logarithm, these expressions have infinitely many periodic values, but direct calculation shows, by taking the principal value of the right side of $(1)$, that they sum is equal to the left side, it's to say, to $\cos (30/\sqrt{2})^\circ$. This means that we can express the cosine of this irrational angle in closed form.

But, as the right hand side of $(1)$ is infinitely-valued, is it correct to use the equality sign between them? Is this a valid representation, if we restrict the right side to take principal values only? Is this rigorous?

I think that is it not trivial to obtain a representation of the trigonometric functions of certain irrational angles as a finite combination of algebraic numbers. Also, not every real number can be so expressed, so I think that there is something interesting in this subject. However, I want to know if this representation is correct.

NOTE: I am reffering to irrational multiples of the form $\alpha \pi$, where $\alpha$ is an algebraic number that can be expressed by radicals. This excludes cases like $\cos \alpha = 1/3$, since, in this case, $\alpha$ cannot be expressed by radicals and is, in fact, a transcendental angle, both in degrees and radians.

• Irrational multiples of $\pi$ can have algebraic, indeed even rational trig functions. For example, let $\theta$ be the angle whose sine is $3/5$ and whose cosine is $4/5$. – André Nicolas May 18 '16 at 15:58
• (Slightly off topic, but it's a nice problem to prove that @AndréNicolas's $\theta$, $\arcsin(3/5)$, is indeed an irrational multiple of $\pi$.) – Akiva Weinberger May 18 '16 at 16:37