Graphs of the Other Four Trigonometric Functions

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node57.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="node59.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node58.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Graphs of the Other Four Trigonometric Functions</H2> <P> As a sidenote, we briefly show the graphs of the other four trig functions, which are <IMG WIDTH=39 HEIGHT=10 ALIGN=BOTTOM ALT="tan x" SRC="img158.gif">, <IMG WIDTH=36 HEIGHT=10 ALIGN=BOTTOM ALT="cot(x)" SRC="img159.gif">, <IMG WIDTH=35 HEIGHT=7 ALIGN=BOTTOM ALT="sec(x)" SRC="img160.gif">, and <IMG WIDTH=35 HEIGHT=7 ALIGN=BOTTOM ALT="csc(x)" SRC="img161.gif">. Again, we build these by simply plotting points and connecting the dots. <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot17.gif" WIDTH=323 HEIGHT=234 ALT="[plot]"> <IMG ALIGN=BOTTOM SRC="plot18.gif" WIDTH=320 HEIGHT=237 ALT="[plot]"><P> <IMG ALIGN=BOTTOM SRC="plot19.gif" WIDTH=321 HEIGHT=234 ALT="[plot]"> <IMG ALIGN=BOTTOM SRC="plot20.gif" WIDTH=322 HEIGHT=236 ALT="[plot]"> </CENTER> <P> In the case of the function <IMG WIDTH=33 HEIGHT=12 ALIGN=BOTTOM ALT="tan(theta)" SRC="img85.gif">, note that the period is <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="pi" SRC="img58.gif"> rather than <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="2 pi" SRC="img122.gif">. Moreover, note that the graph contains infinitely many vertical <B>asymptotes</B>, vertical lines at which the graph is undefined and through which the graph never travels. <P> The same statements can be made about <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cot(theta)" SRC="img162.gif">, although you should note that the location of the vertical asymptotes in <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cot(theta)" SRC="img162.gif"> is different from the location of the vertical asymptotes in <IMG WIDTH=33 HEIGHT=12 ALIGN=BOTTOM ALT="tan(theta)" SRC="img85.gif">. You should note that these vertical asymptotes are located exactly where the function in question is undefined. In the case of <IMG WIDTH=33 HEIGHT=12 ALIGN=BOTTOM ALT="tan(theta)" SRC="img85.gif">, this is exactly where <IMG WIDTH=60 HEIGHT=12 ALIGN=BOTTOM ALT="cos(theta)=0" SRC="img163.gif"> since <P> <CENTER> <IMG WIDTH=97 HEIGHT=34 ALIGN=BOTTOM ALT="tan = sin/cos" SRC="img164.gif"> </CENTER> <P> In the case of <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cot(theta)" SRC="img162.gif">, the vertical asymptotes are located exactly where <IMG WIDTH=58 HEIGHT=12 ALIGN=BOTTOM ALT="sin(theta)=0" SRC="img165.gif"> since <P> <CENTER> <IMG WIDTH=95 HEIGHT=34 ALIGN=BOTTOM ALT="cot = cos/sin" SRC="img166.gif"> </CENTER> <P> Note that the graphs of <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="sec(theta)" SRC="img167.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="csc(theta)" SRC="img168.gif"> live in the range <IMG WIDTH=128 HEIGHT=24 ALIGN=MIDDLE ALT="(-inf,-1]U[1,inf)" SRC="img169.gif">. That is to say, these two functions seem to live exactly where <IMG WIDTH=17 HEIGHT=12 ALIGN=BOTTOM ALT="sin" SRC="img20.gif"> and <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="cos" SRC="img21.gif"> do not. Why is this? The simplest answer is that <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="sec(theta)" SRC="img167.gif"> is the reciprocal of <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos(theta)" SRC="img84.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="csc(theta)" SRC="img168.gif"> is the reciprocal of <IMG WIDTH=29 HEIGHT=12 ALIGN=BOTTOM ALT="sin(theta)" SRC="img86.gif">. Since both <IMG WIDTH=17 HEIGHT=12 ALIGN=BOTTOM ALT="sin" SRC="img20.gif"> and <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="cos" SRC="img21.gif"> are always less than or equal to 1 in absolute value, it must be the case that <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="sec" SRC="img24.gif"> and <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="csc" SRC="img25.gif"> are always greater than or equal to 1 in absolute value (because of the reciprocal relationship that these functions share). <P> As an interesting exercise, you should trace 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="csc(theta)" SRC="img168.gif"> on the same set of axes and note where they intersect and how they relate to one another. Then, the same should be done with <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="cos(theta)" SRC="img84.gif"> and <IMG WIDTH=30 HEIGHT=12 ALIGN=BOTTOM ALT="sec(theta)" SRC="img167.gif">. <P> <HR><center> <table border=1> <td><A HREF="node57.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="node59.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node58.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>