document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=595833>Question 595833</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use an identity to evaluate sin (2 invesre of sin 5/13) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #377362 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=377362\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282%2Asin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"sin%282%2Asin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Broadly speaking this expression is a reference to the sin of twice some angle. This suggests that the identity to use is the one for sin(2x):<BR>\n" );
document.write( "sin(2x) = 2*sin(x)*cos(x)<br><BR>\n" );
document.write( "Applying this pattern to your expression we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"sin%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \"the sine of the angle whose sin is 5/13\". So it obviously has a value of 5/13.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"cos%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \"the cosine of the angle whose sin is 5/13\". Its value is not so obvious. To find it you can use the Pythagorean identity:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1 ALIGN=MIDDLE  ALT=\"sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting in our sin value this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F13%29%5E2+%2B+cos%5E2%28x%29+=+1 ALIGN=MIDDLE  ALT=\"%285%2F13%29%5E2+%2B+cos%5E2%28x%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve for cos(x) we start by simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2F169+%2B+cos%5E2%28x%29+=+1 ALIGN=MIDDLE  ALT=\"25%2F169+%2B+cos%5E2%28x%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtract the fraction:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+=+1+-+25%2F169 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29+=+1+-+25%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+=+169%2F169+-+25%2F169 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29+=+169%2F169+-+25%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+=+144%2F169 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29+=+144%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Find the square root of each side (discarding the negative square root) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+12%2F13 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now that we have values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"sin%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"cos%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can substitute them into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29 ALIGN=MIDDLE  ALT=\"2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%285%2F13%29%2A%2812%2F13%29 ALIGN=MIDDLE  ALT=\"2%2A%285%2F13%29%2A%2812%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2F13%29%2A%2812%2F13%29 ALIGN=MIDDLE  ALT=\"%2810%2F13%29%2A%2812%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%2F169 ALIGN=MIDDLE  ALT=\"120%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This fraction does not reduce so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282%2Asin%5E-1%285%2F13%29%29+=+120%2F169 ALIGN=MIDDLE  ALT=\"sin%282%2Asin%5E-1%285%2F13%29%29+=+120%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );