Trig Functions of Angles Lying on the Axes

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node25.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="trig.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node27.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node26.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H1>Trig Functions of Angles Lying on the Axes</H1> <P> At this point, we need to stop and talk about 4 special angles, which are the ones whose terminal sides lie on the axes. Namely, if <IMG WIDTH=6 HEIGHT=12 ALIGN=BOTTOM ALT="theta" SRC="img1.gif"> is in standard position, then the angles in question are 0°, 90°, 180°, and 270°. In radians, these angles are 0, <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="pi/2" SRC="img57.gif">, <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="pi" SRC="img58.gif">, and <IMG WIDTH=31 HEIGHT=24 ALIGN=MIDDLE ALT="3 pi/2" SRC="img59.gif"> radians. <P> Notice that one cannot build an appropriate right triangle for these 4 angles. This seems to cause problems for finding the trig function values at these angles. However, we can easily remedy this. <P> Note that, in the past, we built the various right triangles in a manner given in the figure below: <P> <CENTER> <IMG SRC="p7-1.gif" WIDTH=213 HEIGHT=142 ALT="[figure]"><P> </CENTER> Then, we see that the trig functions can actually be defined as the following: <P> <CENTER> <IMG WIDTH=190 HEIGHT=128 ALIGN=BOTTOM ALT="[trig defs]" SRC="img60.gif"> </CENTER> <P> Note that these are exactly the same definitions as were given before, except that we are now referring to <I>x</I>, <I>y</I>, and <I>r</I> rather than the adjacent leg, the opposite leg, and the hypotenuse. <P> Given this focus now on <I>x</I>, <I>y</I>, and <I>r</I>, we can now find the trigonometric values at the angles 0°, 90°, 180°, and 270°. We begin with the angle <IMG WIDTH=50 HEIGHT=12 ALIGN=BOTTOM ALT="theta=90 degrees" SRC="img61.gif">. <P> Note in the case <IMG WIDTH=50 HEIGHT=12 ALIGN=BOTTOM ALT="theta=90 degrees" SRC="img61.gif"> that we have <I>x</I>=0, <I>y</I>=1, and <I>r</I>=1. See the diagram below. <P> <CENTER> <IMG SRC="p7-2.gif" WIDTH=213 HEIGHT=142 ALT="[figure]"><P> </CENTER> <P> (Realize that we could let <I>y</I>=2 or 3 or any positive number for that matter. If we let <I>y</I>=2 for example, then we will have <I>r</I>=2 also.) Then, we can find the values of the six trig functions quickly using the definitions mentioned above in terms of <I>x</I>, <I>y</I>, and <I>r</I>. We have <P> <CENTER> <IMG WIDTH=380 HEIGHT=126 ALIGN=BOTTOM ALT="[trig values]" SRC="img62.gif"> </CENTER> <P> Similarly, in the case of <IMG WIDTH=58 HEIGHT=12 ALIGN=BOTTOM ALT="theta=180 degrees" SRC="img63.gif">, we have <I>x</I>=-1, <I>y</I>=0, and <I>r</I>=1. (We will always think of <I>r</I> as the distance from the point (<I>x</I>,<I>y</I>) to the origin (0,0). In that sense then, <I>r</I> will always be nonnegative.) These values then yield the following:<P> <CENTER> <IMG WIDTH=427 HEIGHT=126 ALIGN=BOTTOM ALT="tex2html_wrap946" SRC="img64.gif"> </CENTER> <P> One can easily calculate the values of the trig functions for <IMG WIDTH=42 HEIGHT=12 ALIGN=BOTTOM ALT="theta=0 degrees" SRC="img65.gif"> and <IMG WIDTH=58 HEIGHT=12 ALIGN=BOTTOM ALT="theta=270 degrees" SRC="img66.gif">. We leave this to the reader. <P> To summarize the results of this section, we close with the following table: <P> <CENTER> <TABLE BORDER CELLPADDING=6> <TR VALIGN=BOTTOM> <TD ALIGN=center><IMG WIDTH=93 HEIGHT=21 ALIGN=MIDDLE ALT="" SRC="tables/img1.gif"></TD> <TD ALIGN=center><IMG WIDTH=93 HEIGHT=21 ALIGN=MIDDLE ALT="" SRC="tables/img2.gif"></TD> <TD ALIGN=center><IMG WIDTH=34 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img3.gif"></TD> <TD ALIGN=center><IMG WIDTH=36 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img4.gif"></TD> <TD ALIGN=center><IMG WIDTH=39 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img5.gif"></TD> <TD ALIGN=center><IMG WIDTH=35 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img6.gif"></TD> <TD ALIGN=center><IMG WIDTH=35 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img7.gif"></TD> <TD ALIGN=center><IMG WIDTH=36 HEIGHT=14 ALIGN=BOTTOM ALT="" SRC="tables/img8.gif"></TD> </TR> <TR VALIGN=BOTTOM> <TD ALIGN=center>0°</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>1</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>1</TD> <TD ALIGN=center>und</TD> </TR> <TR VALIGN=BOTTOM> <TD ALIGN=center>90°</TD> <TD ALIGN=center><IMG WIDTH=9 HEIGHT=21 ALIGN=MIDDLE ALT="" SRC="tables/img24.gif"> </TD> <TD ALIGN=center>1</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>1</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>0</TD> </TR> <TR VALIGN=BOTTOM> <TD ALIGN=center>180°</TD> <TD ALIGN=center><IMG WIDTH=9 HEIGHT=9 ALIGN=BOTTOM ALT="" SRC="tables/img26.gif"> </TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>-1</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>-1</TD> <TD ALIGN=center>und</TD> </TR> <TR VALIGN=BOTTOM> <TD ALIGN=center>270°</TD> <TD ALIGN=center><IMG WIDTH=16 HEIGHT=24 ALIGN=MIDDLE ALT="" SRC="tables/img28.gif"> </TD> <TD ALIGN=center>-1</TD> <TD ALIGN=center>0</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>-1</TD> <TD ALIGN=center>und</TD> <TD ALIGN=center>0</TD> </TR> </TABLE> </CENTER> <P> <HR><center> <table border=1> <td><A HREF="node25.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="trig.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node27.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node26.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>