document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118764>Question 1118764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve for 0<=x<=360\r<BR>\n" );
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document.write( "sinx=cos^2x-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734186 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=734186\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with sin(x) = cos^2(x) - 1\r<BR>\n" );
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document.write( "since sin^2(x) + cos^(x) = 1, then cos^2(x) = 1 - sin^2(x).\r<BR>\n" );
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document.write( "equation becomes sin(x) = 1 - sin^2(x) - 1\r<BR>\n" );
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document.write( "simplify to get sin(x) = -sin^2(x)\r<BR>\n" );
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document.write( "add sin^(x) to both sides of the equation to get sin^2(x) + sin(x) = 0\r<BR>\n" );
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document.write( "factor out sin(x) to get sin(x) * (sin(x) + 1) = 0\r<BR>\n" );
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document.write( "solve this quadratic equation to get sin(x) = 0 or sin(x) = -1\r<BR>\n" );
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document.write( "sin(x) = 0 when x = 0 degrees or x = 180 degrees or x = 360 degrees.\r<BR>\n" );
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document.write( "sin(x) = -1 when x = 270 degrees.\r<BR>\n" );
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document.write( "sin(x) = 1 when x = 0, 380, 360 degrees.\r<BR>\n" );
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document.write( "sin(x) is negative when x = 270 degrees.\r<BR>\n" );
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document.write( "those are your solutions.\r<BR>\n" );
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document.write( "you could have found this as follows:\r<BR>\n" );
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document.write( "when sin(x) = 0, x = 0 degrees found by use of your calculator.\r<BR>\n" );
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document.write( "the equivalent angle in the second quadrant is 180 - 0 = 180 degrees.\r<BR>\n" );
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document.write( "the equivalent angle in the third quadrant is 180 + 0 = 180 degrees.\r<BR>\n" );
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document.write( "the equivalent angle in the fourth quadrant is 360 - 0 = 360 degrees.\r<BR>\n" );
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document.write( "that gets you the possible angles of 0, 180, and 360.\r<BR>\n" );
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document.write( "you can use your calculator to confirm that sin(x) = 0 at all of those angles.\r<BR>\n" );
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document.write( "using your calculator, you will find that sin(x) = -1 leads to x = -90 degrees.\r<BR>\n" );
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document.write( "just put arcsin(-1) in your calculator and it will tell you that.\r<BR>\n" );
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document.write( "make sure your calculator is in degree mode before doing it.\r<BR>\n" );
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document.write( "the equivalent positive angle for -90 degrees is found by adding 360 to it.\r<BR>\n" );
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document.write( "you will get x = 270 degrees.\r<BR>\n" );
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document.write( "this angle could be in the third quadrant or it would be in the fourth quadrant.\r<BR>\n" );
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document.write( "it is on the border line between those quadrants.\r<BR>\n" );
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document.write( "if in the fourth quadrant, the equivalent angle in the first quadrant is 360 - 270 = 90 degrees.\r<BR>\n" );
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document.write( "if in the third quadrant, the equivalent angle in the first quadrant is 270 - 180 = 90 degrees.\r<BR>\n" );
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document.write( "either way, the equivalent angle in the first quadrant is 90 degrees.\r<BR>\n" );
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document.write( "the equivalent angle in the second quadrant is 180 - 90 = 90.\r<BR>\n" );
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document.write( "the equivalent angle in the third quadrant is 180 + 90 = 270.\r<BR>\n" );
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document.write( "the equivalent angle in the fouerth quadrant is 360 - 90 = 270.\r<BR>\n" );
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document.write( "either way, the possible angles in the first, second, third, or fourth quadrants is either 90 degrees or 170 degrees.\r<BR>\n" );
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document.write( "equivalent angle means that the trigonometric function gives you the same value except possibly for the sign.\r<BR>\n" );
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document.write( "sin(90) = 1<BR>\n" );
document.write( "sin(270) = -1\r<BR>\n" );
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document.write( "you are looking for the angle between 0 and 360 that has a trig function of -1.\r<BR>\n" );
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document.write( "that has to be 270 degrees.\r<BR>\n" );
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document.write( "the angles between 0 and 360 that have their sine function = 0 or -1 would then be:\r<BR>\n" );
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document.write( "0, 180, 270, 360.\r<BR>\n" );
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document.write( "that's the interval of 0 <= x <= 360, where x is the angle.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "you can confirm that graphically by graphing y = xin(x) and y = cos^2(x) - 1.\r<BR>\n" );
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document.write( "the intersection of those 2 equations is your solution.\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/2018/061801.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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