document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1008893>Question 1008893</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solving the following equation will require you to use the quadratic formula. Solve the equation for &#952; between 0° and 360°, and round your answers to the nearest tenth of a degree.\r<BR>\n" );
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document.write( "2sin^2&#952; = 3&#8722;4cos&#952; </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #624427 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=624427\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> problem is:\r<BR>\n" );
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document.write( "2sin^2(theta) = 3 - 4cos(theta).\r<BR>\n" );
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document.write( "since sin^2(theta) = 1 - cos^2(theta), substitute for sin^2(theta) to get:\r<BR>\n" );
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document.write( "2 * (1 - cos^2(theta) = 3 - 4cos(theta)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "2 - 2cos^2(theta) = 3 - 4cos(theta)\r<BR>\n" );
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document.write( "add 2cos^2(theta) and subtract 2 from both sides of the equation to get:\r<BR>\n" );
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document.write( "0 = 3 - 4cos(theta) + 2cos^2(theta) - 2\r<BR>\n" );
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document.write( "combine like terms and flip the equation to get:\r<BR>\n" );
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document.write( "2cos^2(theta) - 4cos(theta) + 1 = 0\r<BR>\n" );
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document.write( "that's your quadratic equation that you need to factor.\r<BR>\n" );
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document.write( "use the quadratic formula to factor it as follows:\r<BR>\n" );
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document.write( "since the equation is in standard form of ax^2 + bx + c = 0, you get:\r<BR>\n" );
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document.write( "a = 2<BR>\n" );
document.write( "b = -4<BR>\n" );
document.write( "c = 1\r<BR>\n" );
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document.write( "the quadratic formula is:\r<BR>\n" );
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document.write( "cos(theta) = -b plus or minus sqrt(b^2 - 4ac)\r\n" );
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document.write( "                            2a\r\n" );
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document.write( "replace a,b,c with their values and you get:\r<BR>\n" );
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document.write( "cos(theta) = -(-4) plus or minus sqrt((-4)^2 - 4*2*1)\r\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "cos(theta) = (4 + sqrt(8))/4<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "cos(theta) = (4 - sqrt(8))/4\r<BR>\n" );
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document.write( "the result will be:\r<BR>\n" );
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document.write( "cos(theta) = 1.707106781<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "cos(theta) = .2928932188\r<BR>\n" );
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document.write( "cosine can't be greater than 1, so the solution is:\r<BR>\n" );
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document.write( "cos(theta) = .2928932188\r<BR>\n" );
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document.write( "solve for theta to get:\r<BR>\n" );
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document.write( "theta = 72.96875154 degrees.\r<BR>\n" );
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document.write( "that would be in quadrant 1.\r<BR>\n" );
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document.write( "cosine is positive in quadrants 1 and 4 only.\r<BR>\n" );
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document.write( "your angle is therefore in quadrant 1 and in quadrant 4.\r<BR>\n" );
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document.write( "the angle in quadrant 4 is 360 - 72.96875154 = 287.0312485 degrees.\r<BR>\n" );
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document.write( "your solution is that theta = 72.96875154 or 287.0312485.\r<BR>\n" );
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document.write( "round that to a tenth of a degree and your solution is:\r<BR>\n" );
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document.write( "theta = 73.0 or 287.0 degrees.\r<BR>\n" );
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document.write( "the following picture shows the solution graphically.\r<BR>\n" );
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document.write( "2 equations were graphed.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "y = sin^2(theta)\r<BR>\n" );
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document.write( "y = 3 - 4cos(theta)\r<BR>\n" );
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document.write( "the intersection of the 2 equations on the graph is when their values are equal.\r<BR>\n" );
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document.write( "that occurs at theta = 73 and theta = 287 between 0 and 360 degrees.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/120801.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the graphical solution conforming to the algebraic solution is shown below:\r<BR>\n" );
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document.write( "this graph is of the equation y = 2cos^2(theta) - 4cos(theta) + 1\r<BR>\n" );
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document.write( "the solution, in this case, is when the graph crosses the x-axis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/120802.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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