The Graphs of sin(x) and cos(x)

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node45.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node44.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node47.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node46.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>The Graphs of sin <I>x</I> and cos <I>x</I></H2> <P> We are now in a position to graph <nobr>sin <I>x</I></nobr> and <nobr>cos <I>x</I>.</nobr> Literally connecting the dots from a few key values of the functions (from our table above) yields the following: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot1a.gif" WIDTH=321 HEIGHT=237 ALT="[plot]" > <IMG ALIGN=BOTTOM SRC="plot1b.gif" WIDTH=320 HEIGHT=232 ALT="[plot]" > </CENTER> <P> These graphs show one <B>period</B> of the function or one <B>cycle</B> of the graph of each function. From the first remark in the previous section, we can extend these graphs infinitely in both directions to obtain the following: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot2a.gif" WIDTH=321 HEIGHT=235 ALT="[plot]" > <IMG ALIGN=BOTTOM SRC="plot2b.gif" WIDTH=320 HEIGHT=234 ALT="[plot]" > </CENTER> <P> Note also, as we commented in the previous section, that the <I>y</I>-values stay between -1 and 1. <P> Given these graphs, you can probably imagine how these functions could be used to model many different real-world phenomena. We will talk about such examples later in our text. <P> Several natural questions arise now in terms of these graphs. How can we make the period smaller or larger (in essence, to ``stretch'' or ``shrink'' the graph in the horizontal direction)? How can I move the graph up or down? How can I ``stretch'' the graph up, so that the <I>y</I>-values range from <nobr>[-5, 5],</nobr> for example? <P> <HR><center> <table border=1> <td><A HREF="node45.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node44.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node47.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node46.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I><A HREF="http://www.math.psu.edu/sellersj/">James A. Sellers</A> </I> </ADDRESS> </BODY>