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theorem on constructible angles (Theorem)
Theorem   Let % latex2html id marker 354 $ \theta \in \mathbb{R}$. Then the following are equivalent:
  1. An angle of measure % latex2html id marker 356 $ \theta$ is constructible;
  2. % latex2html id marker 358 $ \sin \theta$ is a constructible number;
  3. % latex2html id marker 360 $ \cos \theta$ is a constructible number.
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:
  1. 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$;
  2. If $ x$ 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 $ 1$ from the vertex of the angle. Label the endpoint that is not the vertex of the angle as $ A$.


\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 $ A$ 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 $ 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 % 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 $ 1$ and another leg of length % latex2html id marker 427 $ \tan \theta$ is constructible, which implies that the three listed conditions are true.



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See Also: constructible numbers, compass and straightedge construction, constructible angles with integer values in degrees, exact trigonometry tables, classical problems of constructibility, criterion for constructibility of regular polygon

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Cross-references: right triangle, implies, point, intersects, radius, circle, arc, erect the perpendicular, constructible numbers, triangle, legs, ray, drop the perpendicular, endpoint, length, line segment, interval, periodicity, constructible number, angle, the following are equivalent
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This is version 10 of theorem on constructible angles, born on 2007-06-15, modified 2008-01-13.
Object id is 9605, canonical name is TheoremOnConstructibleAngles.
Accessed 818 times total.

Classification:
AMS MSC12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )
 51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
 33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)

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