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Hometheorem on constructible angles

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# theorem on constructible angles

###### Theorem 1.

Let $\theta\in\mathbb{R}$. Then the following are equivalent:

1. An angle of measure $\theta$ is constructible;

2. $\sin\theta$ is a constructible number;

3. $\cos\theta$ is a constructible number.

###### Proof.

First of all, due to periodicity, we can restrict our attention to the interval $0\leq\theta<2\pi$. Even better, we can further restrict our attention to the interval $0\leq\theta\leq\frac{\pi}{2}$ for the following reasons:

1. If an angle whose measure is $\theta$ is constructible, then so are angles whose measures are $\pi-\theta$, $\pi+\theta$, and $2\pi-\theta$;

2. If $x$ is a constructible number, then so is $|x|$.

If $\theta\in\{0,\frac{\pi}{2}\}$, then clearly an angle of measure $\theta$ is constructible, and $\{\sin\theta,\cos\theta\}=\{0,1\}$. Thus, equivalence has been established in the case that $\theta\in\{0,\frac{\pi}{2}\}$. Therefore, we can restrict our attention even further to the interval $0<\theta<\frac{\pi}{2}$.

Assume that an angle of measure $\theta$ is constructible. Construct such an angle and mark off a line segment of length $1$ from the vertex of the angle. Label the endpoint that is not the vertex of the angle as $A$.

Drop the perpendicular from $A$ to the other ray of the angle. Since the legs of the triangle are of lengths $\sin\theta$ and $\cos\theta$, both of these are constructible numbers.

Now assume that $\sin\theta$ is a constructible number. At one endpoint of a line segment of length $\sin\theta$, erect the perpendicular to the line segment.

From the other endpoint of the given line segment, draw an arc of a circle with radius $1$ so that it intersects the erected perpendicular. Label this point of intersection as $A$. Connect $A$ to the endpoint of the line segment which was used to draw the arc. Then an angle of measure $\theta$ and a line segment of length $\cos\theta$ have been constructed.

A similar procedure can be used given that $\cos\theta$ is a constructible number to prove the other two statements. ∎

Note that, if $\cos\theta\neq 0$, then any of the three statements thus implies that $\tan\theta$ is a constructible number. Moreover, if $\tan\theta$ is constructible, then a right triangle having a leg of length $1$ and another leg of length $\tan\theta$ is constructible, which implies that the three listed conditions are true.

## Mathematics Subject Classification

33B10*no label found*51M15

*no label found*12D15

*no label found*

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