Which angle can't be trisected?

To show that an angle  cannot be trisected, it suffices to show that is not a constructible angle. Let's consider . Then . Now we use the following trigonometric identity:

Let , we have

It can be shown that is irreducible over . Therefore, the degree of over is and hence by the Main Theorem, is not a constructible number. In other words, is not a constructible angle, which in turn implies that cannot be trisected.