Additional Trigonometric Identities

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node61.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node59.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node63.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node62.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Additional Trigonometric Identities</H2> <P> There are several additional trigonometric identities which we want to highlight before moving on to applications of these identities. <P> First, we note the following two identities involving <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"> which follow from the symmetries displayed by the graphs of these functions. They are: <P> <CENTER> <IMG WIDTH=163 HEIGHT=62 ALIGN=BOTTOM ALT="sin(-theta)=-sin(theta), cos(-theta)=cos(theta)" SRC="img186.gif"> </CENTER> <P> These identities now allow us to determine the <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"> of any negative angle in terms of the related absolute value. Hence, we no longer need to ``remember'' any trig values for negative angles. So, for example, <P> <CENTER> <IMG WIDTH=201 HEIGHT=37 ALIGN=BOTTOM ALT="[sin example]" SRC="img187.gif"> </CENTER> <BR> and <BR> <CENTER> <IMG WIDTH=180 HEIGHT=34 ALIGN=BOTTOM ALT="[cos example]" SRC="img188.gif"> </CENTER> <P> That should simplify some of our work tremendously. <P> Next, we want to look at a class of identities known as the <B>sum and difference</B> formulas. They are as follows: <P> <CENTER> <IMG WIDTH=294 HEIGHT=243 ALIGN=BOTTOM ALT="[identities]" SRC="img189.gif"> </CENTER> <P> These are so named because they give us ways to deal with trig values for sums of angles based on the two angles <I>x</I> and <I>y</I> in question. This will prove to be very useful in the examples we give below. <P> There are many related identities that are direct corollaries of the sum and difference formulas mentioned above. We mention some of these here. The following are known as the <B>double-angle formulas</B>: <P> <CENTER> <IMG WIDTH=226 HEIGHT=148 ALIGN=BOTTOM ALT="[formulas]" SRC="img190.gif"> </CENTER> <P> The above are obtained by letting <I>x</I>=<I>A</I> and <I>y</I>=<I>A</I> in the sum formulas previously mentioned. <P> Next, we mention another set of formulas which are again by-products of the sum formulas mentioned above. They are known as the <B>product-to-sum formulas</B>: <P> <CENTER> <IMG WIDTH=320 HEIGHT=122 ALIGN=BOTTOM ALT="[formulas]" SRC="img191.gif"> </CENTER> <P> Finally we mention the <B>half-angle formulas</B>: <P> <CENTER> <IMG WIDTH=227 HEIGHT=159 ALIGN=BOTTOM ALT="[formulas]" SRC="img192.gif"> </CENTER> <P> This is in no way an exhaustive list of the known identities involving trigonometric functions. However, they provide a good foundation or starting point for you in terms of the identities you should know. <P> <HR><center> <table border=1> <td><A HREF="node61.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node59.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node63.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node62.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>