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Revision difference : theorem on constructible angles |
| Version 2 |
Version 1 |
| \PMlinkescapeword{constructible} |
\PMlinkescapeword{constructible} |
| \PMlinkescapeword{label} |
\PMlinkescapeword{label} |
| \PMlinkescapeword{measure} |
\PMlinkescapeword{measure} |
| \PMlinkescapeword{similar} |
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| \PMlinkescapeword{vertex} |
\PMlinkescapeword{vertex} |
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| \begin{thm*} |
\begin{thm*} |
| Let $\theta \in \mathbb{R}$. Then the following are equivalent: |
Let $\theta \in \mathbb{R}$. Then the following are equivalent: |
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| \begin{enumerate} |
\begin{enumerate} |
| \item An angle of \PMlinkname{measure}{AngleMeasure} $\theta$ is \PMlinkname{constructible}{Constructible2}; |
\item An angle of \PMlinkname{measure}{AngleMeasure} $\theta$ is \PMlinkname{constructible}{Constructible2}; |
| \item $\sin \theta$ is a constructible number; |
\item $\sin \theta$ is a constructible number; |
| \item $\cos \theta$ is a constructible number. |
\item $\cos \theta$ is a constructible number. |
| \end{enumerate} |
\end{enumerate} |
| \end{thm*} |
\end{thm*} |
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| \begin{proof} |
\begin{proof} |
| First of all, due to periodicity, we can restrict our attention to the interval $0 \le \theta <2\pi$. Even better, we can further restrict our attention to the interval $0 \le \theta \le \frac{\pi}{2}$ for the following reasons: |
First of all, due to periodicity, we can restrict our attention to the interval $0 \le \theta <2\pi$. Even better, we can further restrict our attention to the interval $0 \le \theta \le \frac{\pi}{2}$ for the following reasons: |
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| \begin{enumerate} |
\begin{enumerate} |
| \item If an angle whose measure is $\theta$ is constructible, then so are angles whose measures are $\pi-\theta$, $\pi+\theta$, and $2\pi-\theta$; |
\item If an angle whose measure is $\theta$ is constructible, then so are angles whose measures are $\pi-\theta$, $\pi+\theta$, and $2\pi-\theta$; |
| \item If $x$ is a constructible number, then so is $|x|$. |
\item If $x$ is a constructible number, then so is $|x|$. |
| \end{enumerate} |
\end{enumerate} |
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If $\theta \in \{0, \frac{\pi}{2} \}$, then clearly an angle of measure $\theta$ is constructible, and $\{\sin \theta, \cos \theta \}=\{0,1\}$. Thus, \PMlinkname{equivalence}{Equivalent3} has been established in the case that $\theta \in \{0,\frac{\pi}{2}\}$. Therefore, we can restrict our attention even further to the interval $0<\theta<\frac{\pi}{2}$.
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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 $0<\theta<\frac{\pi}{2}$.
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| Assume that an angle of measure $\theta$ is constructible. Construct such an angle and mark off a line segment of length 1 from the \PMlinkname{vertex}{Vertex5} of the angle. Label the endpoint that is not the vertex of the angle as $A$. |
Assume that an angle of measure $\theta$ is constructible. Construct such an angle and mark off a line segment of length 1 from the \PMlinkname{vertex}{Vertex5} of the angle. Label the endpoint that is not the vertex of the angle as $A$. |
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|
| \begin{center} |
\begin{center} |
| \begin{pspicture}(-1,-1)(2,3) |
\begin{pspicture}(-1,-1)(2,3) |
| \rput[l](-0.1,0){.} |
\rput[l](-0.1,0){.} |
| \rput[r](2,0){.} |
\rput[r](2,0){.} |
| \rput[a](2,3.464){.} |
\rput[a](2,3.464){.} |
| \psline{->}(0,0)(2,0) |
\psline{->}(0,0)(2,0) |
| \psline{->}(0,0)(2,3.464) |
\psline{->}(0,0)(2,3.464) |
| \psarc(0,0){0.3}{0}{60} |
\psarc(0,0){0.3}{0}{60} |
| \rput[r](0.5,0.3){$\theta$} |
\rput[r](0.5,0.3){$\theta$} |
| \psarc[linecolor=blue](0,0){3}{50}{70} |
\psarc[linecolor=blue](0,0){3}{50}{70} |
| \psdots(0,0)(1.5,2.598) |
\psdots(0,0)(1.5,2.598) |
| \rput[b](1.2,2.3){$A$} |
\rput[b](1.2,2.3){$A$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| 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. |
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. |
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|
| \begin{center} |
\begin{center} |
| \begin{pspicture}(-1,-1)(2,4) |
\begin{pspicture}(-1,-1)(2,4) |
| \rput[l](-0.1,0){.} |
\rput[l](-0.1,0){.} |
| \rput[r](2,0){.} |
\rput[r](2,0){.} |
| \rput[a](2,3.464){.} |
\rput[a](2,3.464){.} |
| \rput[b](1.5,-1){.} |
\rput[b](1.5,-1){.} |
| \psline{->}(0,0)(2,0) |
\psline{->}(0,0)(2,0) |
| \psline{->}(0,0)(2,3.464) |
\psline{->}(0,0)(2,3.464) |
| \psarc(0,0){0.3}{0}{60} |
\psarc(0,0){0.3}{0}{60} |
| \rput[r](0.5,0.3){$\theta$} |
\rput[r](0.5,0.3){$\theta$} |
| \psarc(0,0){3}{50}{70} |
\psarc(0,0){3}{50}{70} |
| \psline[linecolor=blue]{<->}(1.5,3)(1.5,-1) |
\psline[linecolor=blue]{<->}(1.5,3)(1.5,-1) |
| \psdots(0,0)(1.5,2.598)(1.5,0) |
\psdots(0,0)(1.5,2.598)(1.5,0) |
| \rput[b](1.2,2.3){$A$} |
\rput[b](1.2,2.3){$A$} |
| \rput[a](1,-0.3){$\cos \theta$} |
\rput[a](1,-0.3){$\cos \theta$} |
| \rput[l](1.7,1.3){$\sin \theta$} |
\rput[l](1.7,1.3){$\sin \theta$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| 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. |
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. |
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|
| \begin{center} |
\begin{center} |
| \begin{pspicture}(-1,-1)(3,2) |
\begin{pspicture}(-1,-1)(3,2) |
| \rput[l](-0.1,0){.} |
\rput[l](-0.1,0){.} |
| \rput[r](3,0){.} |
\rput[r](3,0){.} |
| \rput[a](2.598,2){.} |
\rput[a](2.598,2){.} |
| \rput[b](2.598,-1){.} |
\rput[b](2.598,-1){.} |
| \psline{->}(0,0)(3,0) |
\psline{->}(0,0)(3,0) |
| \psline[linecolor=blue]{<->}(2.598,-1)(2.598,2) |
\psline[linecolor=blue]{<->}(2.598,-1)(2.598,2) |
| \psdots(0,0)(2.598,0)(2.598,1.5) |
\psdots(0,0)(2.598,0)(2.598,1.5) |
| \rput[b](2.3,1.5){$A$} |
\rput[b](2.3,1.5){$A$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| 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. |
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. |
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|
| \begin{center} |
\begin{center} |
| \begin{pspicture}(-1,-1)(3,2) |
\begin{pspicture}(-1,-1)(3,2) |
| \rput[l](-0.1,0){.} |
\rput[l](-0.1,0){.} |
| \rput[r](3,0){.} |
\rput[r](3,0){.} |
| \rput[a](2.598,2){.} |
\rput[a](2.598,2){.} |
| \rput[b](2.598,-1){.} |
\rput[b](2.598,-1){.} |
| \psline{->}(0,0)(3,0) |
\psline{->}(0,0)(3,0) |
| \psline{<->}(2.598,-1)(2.598,2) |
\psline{<->}(2.598,-1)(2.598,2) |
| \psarc[linecolor=blue](0,0){3}{20}{40} |
\psarc[linecolor=blue](0,0){3}{20}{40} |
| \psline(0,0)(2.598,1.5) |
\psline(0,0)(2.598,1.5) |
| \psarc(2.598,1.5){0.3}{210}{270} |
\psarc(2.598,1.5){0.3}{210}{270} |
| \rput[a](2.4,1){$\theta$} |
\rput[a](2.4,1){$\theta$} |
| \rput[l](2.8,0.8){$\cos \theta$} |
\rput[l](2.8,0.8){$\cos \theta$} |
| \psdots(0,0)(2.598,0)(2.598,1.5) |
\psdots(0,0)(2.598,0)(2.598,1.5) |
| \rput[b](2.3,1.5){$A$} |
\rput[b](2.3,1.5){$A$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| A similar procedure can be used given that $\cos \theta$ is a constructible number to prove the other two statements. |
A similar procedure can be used given that $\cos \theta$ is a constructible number to prove the other two statements. |
| \end{proof} |
\end{proof} |
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| Note that any of the three statements thus implies that $\tan \theta$ is a constructible number; however, proving any of these three statements from the assumption that $\tan \theta$ is a constructible number is a more difficult matter. |
Note that any of the three statements thus implies that $\tan \theta$ is a constructible number; however, proving any of these three statements from the assumption that $\tan \theta$ is a constructible number is a more difficult matter. |
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