Last-Minute Comments Before Seeing The Pictures

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node44.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="node46.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node45.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Last-Minute Comments Before Seeing The Pictures</H2> <P> Two comments are in order before we actually plot these graphs. <P> First, note that the trig functions <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin theta" SRC="img86.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos theta" SRC="img84.gif"> are completely determined for ALL values of <IMG WIDTH=6 HEIGHT=12 ALIGN=BOTTOM ALT="theta" SRC="img1.gif"> once you know <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin theta" SRC="img86.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos theta" SRC="img84.gif"> for <IMG WIDTH=96 HEIGHT=26 ALIGN=MIDDLE ALT="0 .LE. theta .LE. 360" SRC="img118.gif">. This is because a circle spans 360° in one revolution. Thus, for example, the terminal side of the angle 405° lies in exactly the same place as the terminal side of the angle 45° because <nobr>405 = 45 + 360.</nobr> So, <nobr>sin 405° = sin 45°.</nobr> This is also true, for example, for the angles 1400° and 320°. Therefore, if one wishes to calculate the trig value of an angle, one can ``subtract out'' all multiples of 360 from the angle and then study the remaining angle, which is an angle between 0° and 360°. (The same can be said for negative angles, except that one ``adds in'' multiples of 360 rather than subtracting out multiples of 360.) <P> Because of this, the graphs of <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin theta" SRC="img86.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos theta" SRC="img84.gif"> will repeat themselves every 360° or <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="2 pi" SRC="img122.gif"> radians. Indeed, once we have the graphs for <I>x</I> between 0 radians and <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="2 pi" SRC="img122.gif"> radians, we can simply copy this portion forever in both directions to get the complete graph. <P> Second, note that, in the case of <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin theta" SRC="img86.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos theta" SRC="img84.gif">, it will always be the case that <IMG WIDTH=101 HEIGHT=26 ALIGN=MIDDLE ALT="-1 .LE. sin theta .LE. 1" SRC="img123.gif"> and <IMG WIDTH=103 HEIGHT=26 ALIGN=MIDDLE ALT="-1 .LE. cos theta .LE. 1" SRC="img124.gif"> . This can be seen in a variety of ways. The most important item to realize here is that <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="x .LE. r" SRC="img125.gif"> and <IMG WIDTH=36 HEIGHT=22 ALIGN=MIDDLE ALT="y .LE. r" SRC="img126.gif"> where <I>x</I> and <I>y</I> are the lengths of legs of the right triangle in question and <I>r</I> is the length of the hypotenuse. From the Pythagorean Theorem <nobr><I>x</I><sup>2</sup> + <I>y</I><sup>2</sup> = <I>r</I><sup>2 </sup>,</nobr> it should be clear that <I>x</I> and <I>y</I> can never be bigger than <I>r</I>. From a visual point of view, the hypotenuse of a triangle can never be shorter than either of the two legs. (You would not be able to connect the three legs if it were the case that the hypotenuse was shorter than one of the legs.) Hence, we must have <IMG WIDTH=37 HEIGHT=24 ALIGN=MIDDLE ALT="x .LE. r" SRC="img125.gif"> and <IMG WIDTH=36 HEIGHT=22 ALIGN=MIDDLE ALT="y .LE. r" SRC="img126.gif">. <P> Because of this fact, we can see that the graphs of <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin theta" SRC="img86.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos theta" SRC="img84.gif"> must always lie between -1 and 1 in the <I>y</I>-direction. Using a more technical term, we say that the <B>range</B> of the functions is the interval <nobr>[-1, 1].</nobr> <P> <HR><center> <table border=1> <td><A HREF="node44.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="node46.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node45.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>