document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141185>Question 1141185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation 4sin^2(theta) + 2 cos(theta) = 3 for 0 < theta < 2 pi </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #761727 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=761727\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with 4 * sin^2(theta) + 2 * cos(theta) = 3\r<BR>\n" );
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document.write( "since sin^2(theta) + cos^2(theta) = 1, then sin^2(theta) = 1 - cos^2(theta).\r<BR>\n" );
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document.write( "your equation becomes 4 * (1 - cos^2(theta) + 2 * cos(theta) = 3\r<BR>\n" );
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document.write( "simplify to get 4 - 4 * cos^2(theta) + 2 * cos(theta) = 3\r<BR>\n" );
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document.write( "multiply both sides of this equation by -1 to get -4 + 4 * cos^2(theta) - 2 * cos(theta) = -3\r<BR>\n" );
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document.write( "subtract 3 from both sides of this equation to get 4 * cos^2(theta) - 2 * cos(theta) - 1 = 0\r<BR>\n" );
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document.write( "factor this quadratic equation to get cos(theta) = 0.80901699437495 or cos(theta) = -0.30901699437495\r<BR>\n" );
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document.write( "solve for theta to get theta = 36 degrees or theta = 108 degrees.\r<BR>\n" );
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document.write( "you are looking for the answer between 0 and 360 degrees.\r<BR>\n" );
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document.write( "cosine is positive in first and fourth quadrant.\r<BR>\n" );
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document.write( "therefore 36 degrees and 360 - 36 = 324 degrees should be the angles that get you a cosine of 0.80901699437495.\r<BR>\n" );
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document.write( "cosine is negative in the second and third quadrants.\r<BR>\n" );
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document.write( "therefore 108 degrees and 252 degrees should be the angles that get you a cosine of -0.30901699437495.\r<BR>\n" );
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document.write( "note that the reference angle for 108 degrees is 180 - 108 = 72 degrees.\r<BR>\n" );
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document.write( "note also that the equivalent angle in the third quadrant is 180 + 72 degrees = 252 degrees.\r<BR>\n" );
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document.write( "confirm by replacing theta in the original equation with 36 degrees and 324 degrees and with 108 degrees and 252 degrees.\r<BR>\n" );
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document.write( "the original equation should be true with any of these.\r<BR>\n" );
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document.write( "i did the math and confirmed that all angles make the original equation true.\r<BR>\n" );
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document.write( "you can also solve this by graphing.\r<BR>\n" );
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document.write( "the graph looks like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/051801.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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