document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138302>Question 1138302</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help, I have no idea how to do this.\r<BR>\n" );
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document.write( "Find the general solution of sec3x=cosec2x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756131 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=756131\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you know what the trigonometric identify formulas are, you can solve this.\r<BR>\n" );
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document.write( "otherwise, you can solve it by graphing, if you know how to graph them.\r<BR>\n" );
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document.write( "i solved by grphing first, using the desmos.com calculator.\r<BR>\n" );
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document.write( "here's what the calculator shows me.\r<BR>\n" );
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document.write( "in the interval between 0 and 360 degrees, the calculator tells me that the soluton is:\r<BR>\n" );
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document.write( "x = 18, 162, 234, 306.\r<BR>\n" );
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document.write( "subtract 360 from all of those and the calculator tells me that, in the interval from -360 to 0, the solution is:\r<BR>\n" );
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document.write( "x = -54, -126, -198, -342.\r<BR>\n" );
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document.write( "these solutions repeat every 360 degrees, therefore, the general solution would be:\r<BR>\n" );
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document.write( "18 plus or minus k * 360<BR>\n" );
document.write( "162 plus or minus k * 360<BR>\n" );
document.write( "234 plus or minus k * 360<BR>\n" );
document.write( "306 plus or minus k * 360\r<BR>\n" );
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document.write( "k is an integer that is greater than or equal to 0.\r<BR>\n" );
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document.write( "if you wish to solve it algebraically, then it's good to know the trigonometric identities involved.\r<BR>\n" );
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document.write( "a pretty exhaustive list of trigonometric ikdentities can be found at <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle,_triple-angle,_and_half-angle_formulae\" target = \"_blank\">https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle,_triple-angle,_and_half-angle_formulae</a>\r<BR>\n" );
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document.write( "here's a display of the section in the reference that addresses double and triple angle formulas.\r<BR>\n" );
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document.write( "that section is about halfway down in the page of the reference.\r<BR>\n" );
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document.write( "it helps to know that secant is equal to 1 / cosine and that cosecant is equal to 1 divided by sine.\r<BR>\n" );
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document.write( "your equation of secant(3x) = cosecant(2x) becomes:\r<BR>\n" );
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document.write( "1 / cosine(3x) = 1 / sine(2x)\r<BR>\n" );
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document.write( "cross multiply to get sine(2x) = cosine(3x)\r<BR>\n" );
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document.write( "from the identity formulas, you get:\r<BR>\n" );
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document.write( "sine(2x) = 2 * sine(x) * cosine(x)\r<BR>\n" );
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document.write( "cosine(3x) = 4 * cosine^3(x) - 3 * cosine(x)\r<BR>\n" );
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document.write( "sine(2x) = cosine(3x) becomes:\r<BR>\n" );
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document.write( "2 * sine(x) * cosine(x) = 4 * cosine^3(x) - 3 * cosine(x)\r<BR>\n" );
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document.write( "divide both sides of this equation by cosine(x) to get:\r<BR>\n" );
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document.write( "2 * sine(x) = 4 * cosine^2(x) - 3\r<BR>\n" );
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document.write( "since cosine^2(x) is equal to 1 - sine^2(x), the equation becomes:\r<BR>\n" );
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document.write( "2 * sine(x) = 4 * (1 - sine^2(x)) - 3\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "2 * sine(x) = 4 - 4 * sine^2(x) - 3\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "2 * sine(x) = -4 * sine^2(x) + 1\r<BR>\n" );
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document.write( "add 4 * sine^2(x) to both sides of the equation and subtract 1 from both sides of the equation to and order the terms in descending order of degree to get:\r<BR>\n" );
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document.write( "4 * sine^2(x) + 2 * sine(x) - 1 = 0\r<BR>\n" );
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document.write( "if you let y = sine(x), this equation becomes:\r<BR>\n" );
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document.write( "4 * y^2 + 2 * y - 1 = 0\r<BR>\n" );
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document.write( "solve this quadratic equation to get:\r<BR>\n" );
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document.write( "y = -0.80901699437495 or y = 0.30901699437495\r<BR>\n" );
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document.write( "since y = sine(x), you get:\r<BR>\n" );
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document.write( "sine(x) = -0.80901699437495 or sine(x) = 0.30901699437495\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -54 degrees or x = 18 degrees.\r<BR>\n" );
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document.write( "if you add 360 to -54 degrees, it will become the positive angle of 306 degrees.\r<BR>\n" );
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document.write( "the equivalwent angle in the first quadrant would be 360 - 306 = 54 degrees.\r<BR>\n" );
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document.write( "your possible solutions between 0 and 360 degrees would be:\r<BR>\n" );
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document.write( "18 and 54 in the first quadrant.<BR>\n" );
document.write( "180 - 18 = 162 and 180 - 54 = 126 in the second quadrant.<BR>\n" );
document.write( "180 + 18 = 198 and 180 + 54 = 234 in the third quadrant.<BR>\n" );
document.write( "360 - 18 = 342 and 360 - 54 = 306 in the fourth quadrant.\r<BR>\n" );
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document.write( "you would want to evaluate 1 / cosine(3x) and 1 / sine(2x) in each of these angles to see which ones hold true.\r<BR>\n" );
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document.write( "18 is good. *****<BR>\n" );
document.write( "54 is no good.<BR>\n" );
document.write( "162 is good. *****<BR>\n" );
document.write( "126 is no good.<BR>\n" );
document.write( "198 is no good.<BR>\n" );
document.write( "234 is good. *****<BR>\n" );
document.write( "342 is no good.<BR>\n" );
document.write( "306 is good. *****\r<BR>\n" );
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document.write( "the good angles are 18, 162, 234, 306.\r<BR>\n" );
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document.write( "these angles repeat every 360 degrees, therefore the solution is:\r<BR>\n" );
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document.write( "18 plus or minus 360 * k<BR>\n" );
document.write( "162 plus or minus 360 * k<BR>\n" );
document.write( "234 plus or minus 360 * k<BR>\n" );
document.write( "306 plus or minus 360 * k\r<BR>\n" );
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document.write( "this agrees with the graphical solution.\r<BR>\n" );
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document.write( "if you need the answer in radians, just multiply the degrees by pi / 180 and you get the equivalent answer in radians.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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