document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=205115>Question 205115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two cyclists leave the corner of State Street and Main Street simultaneously.  State Street and Main Street are not at right angles;  the cyclists' paths have an angle of  45o  between them.  How far apart are the cyclists after they each travel 7 miles?  Hint:  Use the Law of Cosines\r<BR>\n" );
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document.write( "Can you explain how the law of cosines works and how it would apply to this problem? \r<BR>\n" );
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document.write( "Thank You  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154840 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=154840\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> just about everything you would ever want to know about the law of cosines alnd probably lots more can be found at this web address:<BR>\n" );
document.write( "http://en.wikipedia.org/wiki/Law_of_cosines<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the law of cosines is a generalization of the pythagorean theorem that applies to all triangles regardless if they are right triangles or not.  The pythagorean formula applies to right triangles only.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the pythagorean theorem states:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2+%2B+b%5E2 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2+%2B+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the law of cosines theorem states:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2+%2B+b%5E2+-+2%2Aa%2Ab%2Acos%28c%29 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2+%2B+b%5E2+-+2%2Aa%2Ab%2Acos%28c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your problem states:<BR>\n" );
document.write( "Two cyclists leave the corner of State Street and Main Street simultaneously.  State Street and Main Street are not at right angles;  the cyclists' paths have an angle of  45o  between them.  How far apart are the cyclists after they each travel 7 miles?  Hint:  Use the Law of Cosines<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Your triangle is ABC<BR>\n" );
document.write( "side opposite angle A is side a.<BR>\n" );
document.write( "side opposite angle B is side b.<BR>\n" );
document.write( "side opposite angle C is side c.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since side a and side b are equal, they form an isosceles triangle allowing you to solve this without use of the law of cosines.<BR>\n" );
document.write( "we will, however, solve it first using the law of cosines.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we have:<BR>\n" );
document.write( "side a = 7<BR>\n" );
document.write( "side b = 7<BR>\n" );
document.write( "angle c = 45 degrees.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "law of cosines formula is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2+%2B+b%5E2+-+2%2Aa%2Ab%2Acos%28c%29 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2+%2B+b%5E2+-+2%2Aa%2Ab%2Acos%28c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "substituting known values in that formula we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+7%5E2+%2B+7%5E2+%2B+2%2A7%2A7%2Acos%2845%29 ALIGN=MIDDLE  ALT=\"c%5E2+=+7%5E2+%2B+7%5E2+%2B+2%2A7%2A7%2Acos%2845%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "cos(45) = .707106781...<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+7%5E2+%2B+7%5E2+%2B+2%2A7%2A7%2A%28.707106781%29 ALIGN=MIDDLE  ALT=\"c%5E2+=+7%5E2+%2B+7%5E2+%2B+2%2A7%2A7%2A%28.707106781%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "solving we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+49+%2B+49+%2B+98%2A%28.707106781%29 ALIGN=MIDDLE  ALT=\"c%5E2+=+49+%2B+49+%2B+98%2A%28.707106781%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+28.70353544 ALIGN=MIDDLE  ALT=\"c%5E2+=+28.70353544\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "take square root of both sides to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+5.357568053 ALIGN=MIDDLE  ALT=\"c+=+5.357568053\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "distance between the cyclists is 5.36 miles.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "a picture can be found at the following website:<BR>\n" );
document.write( "http://theo.x10hosting.com/ </SPAN>\n" );
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