PlanetMath: theorem on constructible angles

Proof. First of all, due to periodicity, we can restrict our attention to the interval $% latex2html id marker 365 $ 0 \le \theta <2\pi$$ . Even better, we can further restrict our attention to the interval $% latex2html id marker 367 $ 0 \le \theta \le \frac{\pi}{2}$$ for the following reasons:

If an angle whose measure is $% latex2html id marker 369 $ \theta$$ is constructible, then so are angles whose measures are $% latex2html id marker 371 $ \pi-\theta$$ , $% latex2html id marker 373 $ \pi+\theta$$ , and $% latex2html id marker 375 $ 2\pi-\theta$$ ;
If is a constructible number, then so is $\vert x\vert$ .

If $% latex2html id marker 381 $ \theta \in \{0, \frac{\pi}{2} \}$$ , then clearly an angle of measure $% latex2html id marker 383 $ \theta$$ is constructible, and $% latex2html id marker 385 $ \{\sin \theta, \cos \theta \}=\{0,1\}$$ . Thus, equivalence has been established in the case that $% latex2html id marker 387 $ \theta \in \{0,\frac{\pi}{2}\}$$ . Therefore, we can restrict our attention even further to the interval $% latex2html id marker 389 $ 0<\theta<\frac{\pi}{2}$$ .

Assume that an angle of measure $% latex2html id marker 391 $ \theta$$ is constructible. Construct such an angle and mark off a line segment of length from the vertex of the angle. Label the endpoint that is not the vertex of the angle as .

$\begin{pspicture} % latex2html id marker 86 (-1,-1)(2,3) \rput[l](-0.1,0){.} \rp... ...](0,0){3}{50}{70} \psdots(0,0)(1.5,2.598) \rput[b](1.2,2.3){$A$} \end{pspicture}$

Drop the perpendicular from to the other ray of the angle. Since the legs of the triangle are of lengths $% latex2html id marker 399 $ \sin \theta$$ and $% latex2html id marker 401 $ \cos \theta$$ , both of these are constructible numbers.

$\begin{pspicture} % latex2html id marker 113 (-1,-1)(2,4) \rput[l](-0.1,0){.} \r... ...put[a](0.7,-0.3){$\cos \theta$} \rput[l](1.7,1.3){$\sin \theta$} \end{pspicture}$

Now assume that $% latex2html id marker 403 $ \sin \theta$$ is a constructible number. At one endpoint of a line segment of length $% latex2html id marker 405 $ \sin \theta$$ , erect the perpendicular to the line segment.

$\begin{pspicture}(-1,-1)(3,2) \rput[l](-0.1,0){.} \rput[r](3,0){.} \rput[a](2.59... ...ne[linecolor=blue]{<->}(2.598,-1)(2.598,2) \psdots(0,0)(2.598,0) \end{pspicture}$

From the other endpoint of the given line segment, draw an arc of a circle with radius so that it intersects the erected perpendicular. Label this point of intersection as . Connect to the endpoint of the line segment which was used to draw the arc. Then an angle of measure $% latex2html id marker 413 $ \theta$$ and a line segment of length $% latex2html id marker 415 $ \cos \theta$$ have been constructed.

$\begin{pspicture} % latex2html id marker 156 (-1,-1)(3,2) \rput[l](-0.1,0){.} \r... ...\theta$} \psdots(0,0)(2.598,0)(2.598,1.5) \rput[b](2.3,1.5){$A$} \end{pspicture}$

A similar procedure can be used given that $% latex2html id marker 417 $ \cos \theta$$ is a constructible number to prove the other two statements. $\qedsymbol$

Note that, if $% latex2html id marker 419 $ \cos \theta \neq 0$$ , then any of the three statements thus implies that $% latex2html id marker 421 $ \tan \theta$$ is a constructible number. Moreover, if $% latex2html id marker 423 $ \tan \theta$$ is constructible, then a right triangle having a leg of length

and another leg of length $% latex2html id marker 427 $ \tan \theta$$ is constructible, which implies that the three listed conditions are true.