document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=816780>Question 816780</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use inverse trigonometric functions to find the solutions of the equation that are in the interval [0, 2&#960;).You may enter an exact answer or round your solutions to four decimal places (this is best for this problem). (Enter your 4 answers in a comma separated list.)\r<BR>\n" );
document.write( "\n" );
document.write( "cos(x)(8cos(x) + 4) = 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #491756 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=491756\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%288cos%28x%29+%2B+4%29+=+3 ALIGN=MIDDLE  ALT=\"cos%28x%29%288cos%28x%29+%2B+4%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The left side is factored. But the right side is not zero. So having the left side factored does not help. Multiplying out the left side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8cos%5E2%28x%29+%2B+4cos%28x%29+=+3 ALIGN=MIDDLE  ALT=\"8cos%5E2%28x%29+%2B+4cos%28x%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we get a zero on the right by subtracting 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8cos%5E2%28x%29+%2B+4cos%28x%29+-+3=0 ALIGN=MIDDLE  ALT=\"8cos%5E2%28x%29+%2B+4cos%28x%29+-+3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we factor, if we can. But this will not factor. It is a quadratic however. So we can use the quadratic formula with an \"a\" of 8, a \"b\" of 4 and a \"c\" of -3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-%284%29%2B-sqrt%28%284%29%5E2-4%288%29%28-3%29%29%29%2F2%288%29 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-%284%29%2B-sqrt%28%284%29%5E2-4%288%29%28-3%29%29%29%2F2%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%2816-4%288%29%28-3%29%29%29%2F2%288%29 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%2816-4%288%29%28-3%29%29%29%2F2%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%2816%2B96%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%2816%2B96%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%28112%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%28112%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%284%2A28%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%284%2A28%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%284%2A4%2A7%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%284%2A4%2A7%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-sqrt%284%29%2Asqrt%284%29%2Asqrt%287%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-sqrt%284%29%2Asqrt%284%29%2Asqrt%287%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-2%2A2%2Asqrt%287%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-2%2A2%2Asqrt%287%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-4%2B-4sqrt%287%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-4%2B-4sqrt%287%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%284%28-1%2B-sqrt%287%29%29%29%2F16 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%284%28-1%2B-sqrt%287%29%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-1%2B-sqrt%287%29%29%2F4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-1%2B-sqrt%287%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is short for:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-1%2Bsqrt%287%29%29%2F4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-1%2Bsqrt%287%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-1-sqrt%287%29%29%2F4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-1-sqrt%287%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Using our calculator to get decimal approximations (and rounding them to 4 places):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-1%2B2.6458%29%2F4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-1%2B2.6458%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+%28-1-2.6458%29%2F4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%28-1-2.6458%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+0.4115 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+0.4115\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+-0.9115 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+-0.9115\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Using the inverse cos on our calculator on these decimals... <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+0.4115 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+0.4115\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E-1%280.4115%29 ALIGN=MIDDLE  ALT=\"cos%5E-1%280.4115%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 1.1467. So the reference angle is 1.1467. And since the 0.4115 is positive and since cos is positive in the 1st and 4th quadrants we should get general solution equations of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 1st quadrant)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 4th quadrant)<br><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+-0.9115 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+-0.9115\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E-1%280.9115%29 ALIGN=MIDDLE  ALT=\"cos%5E-1%280.9115%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 0.4239. (Note: Do not enter the minus of -0.9115 when looking for a reference angle! With a minus on the decimal, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E-1%280.9115%29 ALIGN=MIDDLE  ALT=\"cos%5E-1%280.9115%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we will get second quadrant angle, not a reference angle!) So the reference angle is 0.4239. And since the -0.9115 is negative (<i>here</i> is where the minus gets used) and since cos is negative in the 2nd and 3rd quadrants we should get general solution equations of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2B0.4239+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2B0.4239+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 2nd quadrant)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi-0.4239+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi-0.4239+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 3rd quadrant)<BR>\n" );
document.write( "Replacing first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s with 3.1416:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.1416-0.4239+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3.1416-0.4239+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 2nd quadrant)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.1416%2B0.4239+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3.1416%2B0.4239+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 3rd quadrant)<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.7177+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+2.7177+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.5655+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3.5655+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the general solution equations are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.7177+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+2.7177+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.5655+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3.5655+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From these we will try various integer values for n looking for x's that are in the given interval.<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "if n = 0 then x = 1.1467<BR>\n" );
document.write( "if n = 1 (or larger) then x is too large for the interval<BR>\n" );
document.write( "if n = -1 (or smaller) then x is too small for the interval<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.1467+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-1.1467+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "if n = 0 (or smaller) then x is too small for the interval<BR>\n" );
document.write( "if n = 1 then x = 1.1467 + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5.1365<BR>\n" );
document.write( "if n = 2 (or larger) then x is too large for the interval<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.7177+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+2.7177+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "if n = 0 then x = 2.7177<BR>\n" );
document.write( "if n = 1 (or larger) then x is too large for the interval<BR>\n" );
document.write( "if n = -1 (or smaller) then x is too small for the interval<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.5655+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3.5655+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "if n = 0 then x = 3.5655<BR>\n" );
document.write( "if n = 1 (or larger) then x is too large for the interval<BR>\n" );
document.write( "if n = -1 (or smaller) then x is too small for the interval<br><BR>\n" );
document.write( "So the specific solutions within the given interval are:<BR>\n" );
document.write( "1.1467, 5.1365, 2.7177 and 3.5655 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );