| Version 18 |
Version 17 |
| The \PMlinkname{arcsin}{CyclometricFunctions} is the inverse function of the sine --- therefore the composition function |
The \PMlinkname{arcsin}{CyclometricFunctions} is the inverse function of the sine --- therefore the composition function |
| $$f:\,x\mapsto \arcsin(\sin{x})$$ |
$$f:\,x\mapsto \arcsin(\sin{x})$$ |
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is the identity map \, $x\mapsto x$\, on the interval \,
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is the identity map\, $x\mapsto x$\, on the interval\,
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| $[-\frac{\pi}{2},\,\frac{\pi}{2}]$.\, On this interval the \PMlinkescapetext{inner function $\sin$ increases monotonically and continuously from its least value $-1$ to its greatest value 1; then the outer} function $\arcsin$ (i.e. the angle corresponding the sine value) and the whole composition correspondingly grows from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.\, On the next equally long interval\, $[\frac{\pi}{2},\,\frac{3\pi}{2}]$,\, when the inner function decreases from 1 to $-1$, the composition thus decreases from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$,\, evidently again linearly.\, We now have run through a \PMlinkescapetext{period} interval\, |
$[-\frac{\pi}{2},\,\frac{\pi}{2}]$.\, On this interval the \PMlinkescapetext{inner function $\sin$ increases monotonically and continuously from its least value $-1$ to its greatest value 1; then the outer} function $\arcsin$ (i.e. the angle corresponding the sine value) and the whole composition correspondingly grows from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.\, On the next equally long interval\, $[\frac{\pi}{2},\,\frac{3\pi}{2}]$,\, when the inner function decreases from 1 to $-1$, the composition thus decreases from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$,\, evidently again linearly.\, We now have run through a \PMlinkescapetext{period} interval\, |
| $[-\frac{\pi}{2},\,\frac{3\pi}{2}]$\, of the inner function and the composition $f$ and obtained a wedge-formed portion ($\wedge$) of the graph.\, Because of the periodicity the whole graph of $f$ consists of such successive wedges and thus is like a saw \PMlinkescapetext{blade}.\, The {\em triangular-wave function} is continuous.\, Its derivative (away from the singular points\, |
$[-\frac{\pi}{2},\,\frac{3\pi}{2}]$\, of the inner function and the composition $f$ and obtained a wedge-formed portion ($\wedge$) of the graph.\, Because of the periodicity the whole graph of $f$ consists of such successive wedges and thus is like a saw \PMlinkescapetext{blade}.\, The {\em triangular-wave function} is continuous.\, Its derivative (away from the singular points\, |
| $\frac{\pi}{2}+n\pi,\, n\in\mathbb{Z}$) is a |
$\frac{\pi}{2}+n\pi,\, n\in\mathbb{Z}$) is a |
| \PMlinkname{square-wave function}{CommonFourierSeries}. |
\PMlinkname{square-wave function}{CommonFourierSeries}. |
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There are two graphs here so that \PMlinkexternal{pahio}{http://planetmath.org/?op=getuser&id=2872} can choose which one he likes best. |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-6.6,-2.3)(6.7,2.3) |
\begin{pspicture}(-6.6,-2.3)(6.7,2.3) |
| \psline[linecolor=blue](-6.6,-0.3168)(-4.7124,1.5708) |
\psline[linecolor=blue](-6.6,-0.3168)(-4.7124,1.5708) |
| \psline[linecolor=blue](-4.7124,1.5708)(-1.5708,-1.5708) |
\psline[linecolor=blue](-4.7124,1.5708)(-1.5708,-1.5708) |
| \psline[linecolor=blue](-1.5708,-1.5708)(1.5708,1.5708) |
\psline[linecolor=blue](-1.5708,-1.5708)(1.5708,1.5708) |
| \psline[linecolor=blue](1.5708,1.5708)(4.7124,-1.5708) |
\psline[linecolor=blue](1.5708,1.5708)(4.7124,-1.5708) |
| \psline[linecolor=blue](4.7124,-1.5708)(6.6,0.3168) |
\psline[linecolor=blue](4.7124,-1.5708)(6.6,0.3168) |
| \rput[l](-6.6,0){.} |
\rput[l](-6.6,0){.} |
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\psaxes[labels=none,Dx=1.5708,Dy=1.5708]{->}(0,0)(-6.6,-1.8)(7.0,2.3)
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\psaxes[labels=none,Dx=1.5708,Dy=1.5708](0,0)(-6.6,-1.8)(6.7,2.3)
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\rput[a](7.1,-0.25){$x$}
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\rput[a](6.7,-0.4){$x$}
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\rput[r](-0.22,2.35){$y$}
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\rput[r](-0.25,2.3){$y$}
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| \rput[a](0,-2.3){\textbf{Figure:} Graph of $\arcsin(\sin x)$} |
\rput[a](0,-2.3){\textbf{Figure:} Graph of $\arcsin(\sin x)$} |
| \rput[a](-6.2832,-0.4){$2\pi$} |
\rput[a](-6.2832,-0.4){$2\pi$} |
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\rput[a](-4.7124,-0.4){$-\frac{3\pi}{2}$}
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\rput[a](-4.7124,-0.4){$\frac{-3\pi}{2}$}
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| \rput[a](-3.1416,-0.4){$\pi$} |
\rput[a](-3.1416,-0.4){$\pi$} |
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\rput[a](-1.5708,-0.4){$-\frac{\pi}{2}$}
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\rput[a](-1.5708,-0.4){$\frac{-\pi}{2}$}
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| \rput[a](1.5708,-0.4){$\frac{\pi}{2}$} |
\rput[a](1.5708,-0.4){$\frac{\pi}{2}$} |
| \rput[a](3.1416,-0.4){$\pi$} |
\rput[a](3.1416,-0.4){$\pi$} |
| \rput[a](4.7124,-0.4){$\frac{3\pi}{2}$} |
\rput[a](4.7124,-0.4){$\frac{3\pi}{2}$} |
| \rput[a](6.2832,-0.4){$2\pi$} |
\rput[a](6.2832,-0.4){$2\pi$} |
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\rput[r](-0.3,-1.5708){$-\frac{\pi}{2}$}
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\rput[r](-0.4,-1.5708){$\frac{-\pi}{2}$}
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\rput[r](-0.3,1.5708){$\frac{\pi}{2}$}
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\rput[r](-0.4,1.5708){$\frac{\pi}{2}$}
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| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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\begin{figure}[!htb] |
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\begin{center} |
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\includegraphics{triangular-wave.eps} |
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\end{center} |
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\caption{Graph of $\arcsin(\sin x)$} |
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\end{figure} |
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| Sometimes, such a function is called {\em saw-tooth function}, although this name usually refers to a discontinuous function with graph consisting of ascending ($/$) or descending ($\backslash$) line segments with jumps. |
Sometimes, such a function is called {\em saw-tooth function}, although this name usually refers to a discontinuous function with graph consisting of ascending ($/$) or descending ($\backslash$) line segments with jumps. |
