document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1002887>Question 1002887</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have tried so many times to solve the problem\r<BR>\n" );
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document.write( "Find the exact value of<BR>\n" );
document.write( "sin (x/2) = 2cos^2(x)-1\r<BR>\n" );
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document.write( "I've changed the sin (x/2) to the half angle identity and changed the 2cos^2(x)-1 to cos 2x, which is the double angle identity and tried to solve to be able to set everything to 0, but nothing I have tried worked. Please help as best as you can.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619753 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=619753\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of<BR>\n" );
document.write( "sin (x/2) = 2cos^2(x)-1<BR>\n" );
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document.write( "The trig functions have to have the same argument.<BR>\n" );
document.write( "sin^2(x/2) = (1 - cos(x))/2<BR>\n" );
document.write( "----<BR>\n" );
document.write( "sin(x/2) = 2cos^2(x)-1<BR>\n" );
document.write( "Square both sides<BR>\n" );
document.write( "sin^2(x/2) = 4cos^4(x) - 4cos^2(x) + 1<BR>\n" );
document.write( "(1 - cos(x))/2 = 4cos^4(x) - 4cos^2(x) + 1<BR>\n" );
document.write( "1 - cos(x) = 8cos^4(x) - 8cos^2(x) + 2<BR>\n" );
document.write( "8cos^4(x) - 8cos^2(x) + cos(x) + 1 = 0<BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "If you inspect<BR>\n" );
document.write( "8x^4 - 8x^2 + x + 1 = 0 it's obvious that x = -1 is a zero.<BR>\n" );
document.write( "--> cos(x) = -1<BR>\n" );
document.write( "--> x = pi + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "Divide the quartic by (x+1)<BR>\n" );
document.write( "--> 8x^3 - 8x^2 + 1 = 0<BR>\n" );
document.write( "It's less obvious, x = 1/2 is a zero.<BR>\n" );
document.write( "--> cos(x) = 0.5<BR>\n" );
document.write( "--> x = pi/3 + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "and<BR>\n" );
document.write( "--> x = 2pi/3 + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "Divide the cubic by (x - 1/2):<BR>\n" );
document.write( "--> 8x^2 - 4x - 2 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2%2B-4x%2B-2+=+0 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-4x%2B-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-4%29%5E2-4%2A8%2A-2=80 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-4%29%5E2-4%2A8%2A-2=80\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=80 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--4%2B-sqrt%28+80+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--4%2B-sqrt%28+80+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+80+%29%29%2F2%5C8+=+0.809016994374947 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+80+%29%29%2F2%5C8+=+0.809016994374947\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-4%29-sqrt%28+80+%29%29%2F2%5C8+=+-0.309016994374947 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-4%29-sqrt%28+80+%29%29%2F2%5C8+=+-0.309016994374947\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2%2B-4x%2B-2 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-4x%2B-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2%2B-4x%2B-2+=+%28x-0.809016994374947%29%2A%28x--0.309016994374947%29 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-4x%2B-2+=+%28x-0.809016994374947%29%2A%28x--0.309016994374947%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.809016994374947, -0.309016994374947.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B-4%2Ax%2B-2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B-4%2Ax%2B-2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "======================<BR>\n" );
document.write( "--> cos(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4+%2B-sqrt%285%29%2F4 ALIGN=MIDDLE  ALT=\"1%2F4+%2B-sqrt%285%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--> x = pi/5 + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "--> x = 9pi/5 + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "and x = 3pi/5 + n*2pi, n = ±0,1,2,3...<BR>\n" );
document.write( "and x = 7pi/5 + n*2pi, n = ±0,1,2,3...\r<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );