Determining The Exact Value of sin(pi/12)

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node63.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="node65.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node64.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=63 HEIGHT=16 ALIGN=ABSMIDDLE ALT="sin(pi/12)" SRC="img193.gif"></H3> <P> We want to determine the exact value of <IMG WIDTH=63 HEIGHT=16 ALIGN=ABSMIDDLE ALT="sin(pi/12)" SRC="img193.gif"> using known trig values. Note that <P> <CENTER> <IMG WIDTH=92 HEIGHT=30 ALIGN=BOTTOM ALT="pi/12 = pi/3 - pi/4" SRC="img194.gif"><P> </CENTER> <P> (In terms of degrees, we have noted that <nobr>15° = 60° - 45°).</nobr> Then, from the sum formula for <IMG WIDTH=17 HEIGHT=12 ALIGN=BOTTOM ALT="sin" SRC="img20.gif"> given above, we know that <P> <CENTER> <IMG WIDTH=362 HEIGHT=201 ALIGN=BOTTOM ALT="eqnarray628" SRC="img196.gif"> </CENTER> <P> It would be extremely difficult to find this <B>exact</B> value without the use of this difference formula. A problem similar to this one which would involve a sum formula would be to calculate the exact value of <nobr>sin 75°.</nobr> <P> <HR><center> <table border=1> <td><A HREF="node63.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="node65.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node64.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>