document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1119527>Question 1119527</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation in the interval [0,\, 2  \pi]. If there is more than one solution write them separated by commas. \r<BR>\n" );
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document.write( "Hint: To solve this problem you will need the quadratic formula, inverse trigonometric functions and the symmetry of the unit circle.<BR>\n" );
document.write( "(\cos x)^2 - 1.3 \cos x - 1.98  =0\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735094 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=735094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the problem is:\r<BR>\n" );
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document.write( "cos^2(x) - 1.3 * cos(x) - 1.98 = 0\r<BR>\n" );
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document.write( "let y = cos(x).\r<BR>\n" );
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document.write( "the problem becomes:\r<BR>\n" );
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document.write( "y^2 - 1.3 * y - 1.98 = 0\r<BR>\n" );
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document.write( "use the quadratic formula to get:\r<BR>\n" );
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document.write( "y = -.9 or y = 2.2\r<BR>\n" );
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document.write( "since y = cos(x), you get:\r<BR>\n" );
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document.write( "cos(x) = -.9 or cos(x) = 2.2\r<BR>\n" );
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document.write( "working this problem in radians, and solving for the angle in the first quadrant, where all trig functions are positive, you get:\r<BR>\n" );
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document.write( "arccos(.9) = .4510268118 or arccos(2.2) = undefined, since cosine function has a maximum value of 1.\r<BR>\n" );
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document.write( "your solution in the first quadrant is that the angle is .4510268118 radians.\r<BR>\n" );
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document.write( "your cosine function is negative, however,.\r<BR>\n" );
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document.write( "that means the angle can only be in the second quadrant or the third quadrant.\r<BR>\n" );
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document.write( "the equivalent angle in the second quadrant is x = pi - .4510268118 = 2.690565842 radians.\r<BR>\n" );
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document.write( "the equivalent angle in the third quadrant is x = pi + .4510268118 = 3.592619465 radians.\r<BR>\n" );
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document.write( "if you graph the function of cos^2(x) - 1.3 * cos(x) - 1.98, you will see that the value of the function is 0 when x = 2.690565842 and when x = 3.592619465.\r<BR>\n" );
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document.write( "round these values to 3 decimal places, and they become x = 2.691 radians and x = 3.593 radians as shown on the graph.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/070701.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the unshaded area of this graph goes from x = 0 to x = 2 * pi.\r<BR>\n" );
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document.write( "that's the interval for the value of x that you want.\r<BR>\n" );
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document.write( "the maximum value of cos(x) is plus or minus 1, and the maximum value of cos^2(x) is also plus or minus 1.   \r<BR>\n" );
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document.write( "note that 2 * pi is equal to 2 * 3.141592654 which is equal to 6.283185307.\r<BR>\n" );
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document.write( "the unshaded area of the graph therefore goes from x = 0 to x = approximately 6.283, when rounding to 3 decimal places.\r<BR>\n" );
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