Determining The Exact Value of cos(22 1/2 degrees)

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node64.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node63.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node66.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node65.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H3>Determining The Exact Value of <IMG WIDTH=60 HEIGHT=46 ALIGN=MIDDLE ALT="cos(22 1/2 degrees)" SRC="img199.gif"></H3> <P> As a second example of using these identities, we want to calculate <IMG WIDTH=56 HEIGHT=21 ALIGN=ABSMIDDLE ALT="cos(22 1/2 degrees)" SRC="img198.gif">. Again, this is a very difficult problem if you do not have the identities at your disposal (or if you do not know which one of the identities to use). <P> Here we notice that <IMG WIDTH=83 HEIGHT=35 ALIGN=ABSMIDDLE ALT="22 1/2 = 45/2" SRC="img200.gif">. Hence, we can apply the appropriate half-angle formula to obtain the following: <P> <center> <IMG WIDTH=213 HEIGHT=195 ALIGN=BOTTOM ALT="[calculation]" SRC="img201.gif"> </center> <P> We know in this case to choose the positive sign, since the angle in question ( <IMG WIDTH=31 HEIGHT=31 ALIGN=MIDDLE ALT="22 1/2 degrees" SRC="img202.gif"> ) lies in Quadrant I and <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="cos" SRC="img21.gif"> is positive in Quadrant I. Hence, our answer is <P> <center> <IMG WIDTH=157 HEIGHT=50 ALIGN=BOTTOM ALT="[result]" SRC="img203.gif"> </center> <P> Again, note that finding this exact value is nearly impossible without this identity. <P> <HR><center> <table border=1> <td><A HREF="node64.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node63.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node66.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node65.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>