document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1018648>Question 1018648</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact solutions for 2sinx-1=cscx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #634719 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=634719\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation is:\r<BR>\n" );
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document.write( "2sin(x) - 1 = csc(x)\r<BR>\n" );
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document.write( "since csc(x) = 1/sin(x), this equation becomes:\r<BR>\n" );
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document.write( "2sin(x) - 1 = 1/sin(x)\r<BR>\n" );
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document.write( "multiply both sides of this equation by sin(x) to get:\r<BR>\n" );
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document.write( "sin(x) * (2 * sin(x) - 1) = 1\r<BR>\n" );
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document.write( "distribute the multiplication to get:\r<BR>\n" );
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document.write( "2 * sin^2(x) - sin(x) = 1\r<BR>\n" );
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document.write( "subtract 1 from both sides of the equation to get:\r<BR>\n" );
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document.write( "2 * sin^2(x) - sin(x) - 1 = 0\r<BR>\n" );
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document.write( "this is a quadratic equation that you can solve by using the quadratic formula.\r<BR>\n" );
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document.write( "you will get sin(x) = 1 or sin(x) = -1/2.\r<BR>\n" );
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document.write( "since csc(x) = 1/sin(x), then csc(x) = 1 or csc(x) = -2.\r<BR>\n" );
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document.write( "when sin(x) = 1, then csc(x) - 1, and the equation of 2 * sin(x) - 1 = csc(x) becomes 2 * 1 - 1 = 1.<BR>\n" );
document.write( "simplify this to get 2-1 = 1 which results in 1 = 1.<BR>\n" );
document.write( "this confirms that sin(x) = 1 is a solution to the equation.\r<BR>\n" );
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document.write( "when sin(x) = -1/2, then csc(x) = -2, and the equation of 2 * sin(x) - 1 = csc(x) becomes 2 * -1/2 - 1 = -2 which becomes -1 - 1 = -2 which results in -2 = -2.<BR>\n" );
document.write( "this confirms that sin(x) = -1/2 is also a solution to the equation.\r<BR>\n" );
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document.write( "your solution is therefore:\r<BR>\n" );
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document.write( "sin(x) = 1 or sin(x) = -1/2.\r<BR>\n" );
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document.write( "when sin(x) = 1, x = 90 degrees.\r<BR>\n" );
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document.write( "when sin(x) = -1/2, your calculator will tell you that x is equal to -30 degrees.\r<BR>\n" );
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document.write( "-30 degrees is in quadrant 4.\r<BR>\n" );
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document.write( "convert that to a positive angle by adding 360 to it and you will get x = 330 degrees.\r<BR>\n" );
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document.write( "the sin is negative in quadrant 3 and quadrant 4.\r<BR>\n" );
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document.write( "your reference angle is 30 degrees, which is the equivalent angle in the first quadrant.\r<BR>\n" );
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document.write( "your reference angle becomes 180 + 30 = 210 degrees in quadrant 3.\r<BR>\n" );
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document.write( "your solution is that:\r<BR>\n" );
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document.write( "sin(x) = 1 or sin(x) = -1/2.\r<BR>\n" );
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document.write( "x = 90 degrees or 210 degrees or 330 degrees.\r<BR>\n" );
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document.write( "since it was not specified during what interval, then your solution is:\r<BR>\n" );
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document.write( "sin(x) = 1 or sin(x) = -1/2\r<BR>\n" );
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document.write( "this also implies csc(x) = 1 or csc(x) = -2.\r<BR>\n" );
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document.write( "your angle will be:\r<BR>\n" );
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document.write( "90 degrees plus or minus k * 360.<BR>\n" );
document.write( "210 degrees plus or minus k * 360.<BR>\n" );
document.write( "330 degrees plus or minus k * 360.\r<BR>\n" );
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document.write( "k is a positive integer from 1 to infinity.\r<BR>\n" );
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document.write( "the graph of your equation will be shown below.\r<BR>\n" );
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document.write( "there will be 4 graphs.\r<BR>\n" );
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document.write( "1 close in view to show you the details between 0 and 360 degrees.\r<BR>\n" );
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document.write( "2 far out views to show you the details between -720 and 720 degrees.<BR>\n" );
document.write( "both graphs are the same, but with different intersections shown because the intersections were too close to show them all in 1 graph.\r<BR>\n" );
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document.write( "1 far far out view that goes further than -720 to 720 degrees but without the details.<BR>\n" );
document.write( "this graph will have a horizontal black line at y = 1 and y = -2 to show you where the intersection points are.<BR>\n" );
document.write( "this graph shows you that the possible angles for the solution go on indefinitely in both directions.\r<BR>\n" );
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document.write( "here you go:\r<BR>\n" );
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document.write( "close in view.....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/020803.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "far out view 1....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/020804.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "far out view 2....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/020805.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "far far out view....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/020806.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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