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Version 12 |
| \PMlinkescapeword{axis} |
\PMlinkescapeword{axis} |
| \PMlinkescapeword{right} |
\PMlinkescapeword{right} |
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By the {\em slope angle} of a line in the plane $\mathbb{R}^2$ is sometimes meant the angle measured from the positive $x$-axis to the line in the counterclockwise direction. This procedure is unconvenient in the respect that it determines for the descending lines an obtuse slope angle, which is not directly obtainable e.g. from a calculator. A more convenient definition is as follows.
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By the {\em slope angle} of a line in the plane $\mathbb{R}^2$ is sometimes meant the angle measused from the positive $x$-axis to the linein the counterclocwise direction. This procedure is unconvenient in the respect that it determines for the descending lines an obtuse slope angle, which is not directly obtainable e.g. from a calculator. A more convenient definition is as follows.
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| The {\em slope angle} $\alpha$ of any line in $\mathbb{R}^2$ is the \PMlinkname{angle between}{AngleBetweenTwoLines} the line and $x$-axis, equipped with the minus sign in the case that the line is descending. |
The {\em slope angle} $\alpha$ of any line in $\mathbb{R}^2$ is the \PMlinkname{angle between}{AngleBetweenTwoLines} the line and $x$-axis, equipped with the minus sign in the case that the line is descending. |
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| This definition gives the slope angle for all lines of the plane, satisfying |
This definition gives the slope angle for all lines of the plane, satisfying |
| $$-90^\circ < \alpha \le 90^\circ.$$ |
$$-90^\circ < \alpha \le 90^\circ.$$ |
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| For all non-vertical lines having a slope $m$, the slope angle is obtainable from the \PMlinkescapetext{formula} |
For all non-vertical lines having a slope $m$, the slope angle is obtainable from the \PMlinkescapetext{formula} |
| $$\alpha = \arctan{m};$$ |
$$\alpha = \arctan{m};$$ |
| conversely |
conversely |
| $$m = \tan\alpha.$$ |
$$m = \tan\alpha.$$ |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-7,-3)(7,3) |
\begin{pspicture}(-7,-3)(7,3) |
| \psline[linecolor=red]{->}(-7,0)(-1,0) |
\psline[linecolor=red]{->}(-7,0)(-1,0) |
| \psline[linecolor=red]{->}(1,0)(7,0) |
\psline[linecolor=red]{->}(1,0)(7,0) |
| \psline[linecolor=blue](-7,-2)(-1,2) |
\psline[linecolor=blue](-7,-2)(-1,2) |
| \psline[linecolor=blue](1,3)(7,-3) |
\psline[linecolor=blue](1,3)(7,-3) |
| \psarc(-4,0){0.3}{0}{33.69} |
\psarc(-4,0){0.3}{0}{33.69} |
| \psarc(4,0){0.3}{-40.5}{0} |
\psarc(4,0){0.3}{-40.5}{0} |
| \rput[r](-3.2,0.2){$\alpha$} |
\rput[r](-3.2,0.2){$\alpha$} |
| \rput[r](5.0,-0.3){$|\alpha|$} |
\rput[r](5.0,-0.3){$|\alpha|$} |
| \rput[r](-0.8,-0.2){$x$} |
\rput[r](-0.8,-0.2){$x$} |
| \rput[r](7.15,-0.2){$x$} |
\rput[r](7.15,-0.2){$x$} |
| \rput(-3.5,-2.5){$\mbox{Positive \,}\alpha$} |
\rput(-3.5,-2.5){$\mbox{Positive \,}\alpha$} |
| \rput(3.5,-2.5){$\mbox{Negative \,}\alpha$} |
\rput(3.5,-2.5){$\mbox{Negative \,}\alpha$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{mcgrawhill} ``Slope angle.'' \emph{McGraw-Hill Dictionary of Scientific and Technical Terms.} McGraw-Hill Companies, Inc., 2003. Accessed via Answers.com on 7 June 2007. URL: \PMlinkexternal{http://www.answers.com/topic/slope-angle}{http://www.answers.com/topic/slope-angle} |
\bibitem{mcgrawhill} ``Slope angle.'' \emph{McGraw-Hill Dictionary of Scientific and Technical Terms.} McGraw-Hill Companies, Inc., 2003. Accessed via Answers.com on 07 June 2007. URL: \PMlinkexternal{http://www.answers.com/topic/slope-angle}{http://www.answers.com/topic/slope-angle} |
| \bibitem{1728}``Slope, Distance and Equation Calculator.'' {\em 1729 Software Systems.} Accessed on 24 June 2007. |
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| URL: http://www.1728.com/distance.htm |
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| \end{thebibliography} |
\end{thebibliography} |
