document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168917>Question 1168917</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)<BR>\n" );
document.write( "8 sec(x) csc(x) = 16 csc(x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #793571 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=793571\">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( "8*sec(x) * csc(x) = 16*csc(x)<BR>\n" );
document.write( "sec(x) = 1/cos(x)<BR>\n" );
document.write( "csc(x) = 1/sin(x)<BR>\n" );
document.write( "equation becomes:<BR>\n" );
document.write( "8*1/cos(x)*1/sin(x) = 16/sin(x)<BR>\n" );
document.write( "simplify  to get:<BR>\n" );
document.write( "8/cos(x)*1/sin(x) = 16/sin(x)<BR>\n" );
document.write( "multiply both sides of the equation by sin(x) to get:<BR>\n" );
document.write( "8/cos(x) = 16*sin(x)/sin(x)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "8/cos(x) = 16<BR>\n" );
document.write( "multiply both sides of this equation by cos(x) and divide both sides of this equation by 16 to get:<BR>\n" );
document.write( "8/16 = cos(x)<BR>\n" );
document.write( "solve for cos(x) to get:<BR>\n" );
document.write( "cos(x) = 1/2<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = arccos(1/2) = 60 degrees.<BR>\n" );
document.write( "sec(x) = 1/cos(x) = 1/cos(60) = 2<BR>\n" );
document.write( "csc(x) = 1/sin(x) = 1/sin(60) = 1.154700538<BR>\n" );
document.write( "your equation of:<BR>\n" );
document.write( "8*sec(x)*csc(x) = 16*csc(x) becomes:<BR>\n" );
document.write( "8*2*1.154700538 = 16*1.154700538<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "16*1.154700538 = 16*1.154700538<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "18.47520861 = 18.47520861<BR>\n" );
document.write( "this confirms the solution is correct.<BR>\n" );
document.write( "the solution is x = 60 degrees.\r<BR>\n" );
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document.write( "sine and cosine are both positive in the first quadrant.<BR>\n" );
document.write( "sine is positive and cosine is negative in the second quadrant.<BR>\n" );
document.write( "sine and cosine are both negative in the third quadrant.<BR>\n" );
document.write( "sine is negative and cosine is positive in the fourth quadrant.\r<BR>\n" );
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document.write( "to test the solution, we need to look at all 4 quadrants to see if the solution is valid in any quadrant other than the first quadrant.<BR>\n" );
document.write( "to confirm, we'll look at all 4 quadrants.\r<BR>\n" );
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document.write( "the equations we'll use are:<BR>\n" );
document.write( "y = 16*csc(x) which we'll convert to 16/sin(x)<BR>\n" );
document.write( "y = 8*sec(x)*csc(x) which we'll convert to 8/(cos(x)*sin(x))\r<BR>\n" );
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document.write( "60 degrees in the second quadrant is equal to 180 - 60 = 120 degrees.<BR>\n" );
document.write( "60 degrees in the third quadrant is equal to 180 + 60 = 240 degrees.<BR>\n" );
document.write( "60 degrees in the fourth quadrant is equal to 360 - 60 = 300 degrees.\r<BR>\n" );
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document.write( "we'll use the calculator to get the results.\r<BR>\n" );
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document.write( "16/sin(60) = 18.47520861.<BR>\n" );
document.write( "8/(cos(60)*sin(69)) = 18.47520861<BR>\n" );
document.write( "quadrant 1 is good. *****\r<BR>\n" );
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document.write( "16/sin(120) = 18.47520861<BR>\n" );
document.write( "8/(cos(120)*sin(120)) =  -18.47520861.<BR>\n" );
document.write( "quadrant 2 is not good.\r<BR>\n" );
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document.write( "16/sin(240) = -18.47520861<BR>\n" );
document.write( "8/(cos(240)*sin(240)) = 18.47520861<BR>\n" );
document.write( "quadrant 3 is not good.\r<BR>\n" );
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document.write( "16/sin(300) = -18.47520861<BR>\n" );
document.write( "8/(cos(300)*sin(300)) = -18.47520861<BR>\n" );
document.write( "quadrant 4 is good. *****\r<BR>\n" );
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document.write( "it looks like the solution is good in the first and fourth quadrant.\r<BR>\n" );
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document.write( "in the first quadrant, they have the same value and are both positive, making them equal to each other.<BR>\n" );
document.write( "in the fourth quadrant, they have the same value and are both negative, making them equal to each other.\r<BR>\n" );
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document.write( "in the second and third quadrant, they have the same value, but have opposite signs, therefore not equal.\r<BR>\n" );
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document.write( "we'll confirm further through the use of a graph, where it should be able to see when they are equal.<BR>\n" );
document.write( "the graph of y = 16*csc(x) and the graph of y = 8*sec(x)*csc(x) will have the same value when they intersect.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "your solution in degrees is that the equation is true when x = 60 and x = 300. degrees.\r<BR>\n" );
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document.write( "radians = degrees * pi / 180.\r<BR>\n" );
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document.write( "60 degrees = 60 * pi / 180 = pi/3.\r<BR>\n" );
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document.write( "300 degrees = 300 * pi / 180 = 5pi/3.\r<BR>\n" );
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document.write( "your solution in radians is that the equation is true when x = pi/3 and x = 5pi/3.\r<BR>\n" );
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document.write( "if i had thought about it a little clearer, i might have been able to figure out what quadrant the answer would be in without testing all 4 quadrants.\r<BR>\n" );
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document.write( "in the first quadrant, sine and cosine are both positive, so sine is positive and sine * cosine is positive, making the solution good in the first quadrant.\r<BR>\n" );
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document.write( "in the second quadrant, sine is positive and cosine is negative, so sine is positive and sine * cosine is negative, making the solution not good in the second quadrant.\r<BR>\n" );
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document.write( "in the third quadrant, sine is negative and cosine is negative, so sine is negative and sine * cosine is positive, making the solution not good in the third quadrant.\r<BR>\n" );
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document.write( "in the fourth quadrant, sine is negative and cosine is positive, so sine is negative and sine * cosine is negative, making the solution good in the fourth quadrant\r<BR>\n" );
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document.write( " </SPAN>\n" );
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