document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151636>Question 1151636</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following trigonometric equation:<BR>\n" );
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document.write( "A) It has the same solutions as the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%28sin%28x%29%2Bcos%28x%29%29%5E2-3%28sin%28x%29%2Bcos%28x%29%29%2B2 ALIGN=MIDDLE  ALT=\"0=%28sin%28x%29%2Bcos%28x%29%29%5E2-3%28sin%28x%29%2Bcos%28x%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B) the equation has no real solutions<BR>\n" );
document.write( "C) It has the same solutions as the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%28sqrt%282%29sin%28x%2Bpi%2F4%29-2%29%28sqrt%282%29sin%28x%2Bpi%2F4%29-1%29 ALIGN=MIDDLE  ALT=\"0=%28sqrt%282%29sin%28x%2Bpi%2F4%29-2%29%28sqrt%282%29sin%28x%2Bpi%2F4%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D) It has the same solutions as the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%5E2-3%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%2B2 ALIGN=MIDDLE  ALT=\"0=%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%5E2-3%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "E) none of these<BR>\n" );
document.write( "Note: can you please show me how to find out whether the equation has a real solution or not. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773493 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=773493\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%5E2+=+A%5E2+%2B+2%2AA%2AB+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%5E2+=+A%5E2+%2B+2%2AA%2AB+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+sin%5E2%28x%29+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+cos%5E2%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in A = sin(x) and B = cos(x)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+sin%5E2%28x%29+%2B+cos%5E2%28x%29+%2B+2%2Asin%28x%29%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+highlight%28sin%5E2%28x%29+%2B+cos%5E2%28x%29%29+%2B+2%2Asin%28x%29%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The portion in red\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+highlight%281%29+%2B+2%2Asin%28x%29%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"%28sin%28x%29%2Bcos%28x%29%29%5E2+=+highlight%281%29+%2B+2%2Asin%28x%29%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to 1 due to the pythagorean trig identity\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+%2B+C+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+C ALIGN=MIDDLE  ALT=\"%28sin%28x%29%2Bcos%28x%29%29%5E2+%2B+C+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add C to both sides\r<BR>\n" );
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document.write( "We'll let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 ALIGN=MIDDLE  ALT=\"C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 ALIGN=MIDDLE  ALT=\"C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29+-+3cos%28x%29-3sin%28x%29%2B2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2+=+2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms and do a bit of rearranging.\r<BR>\n" );
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document.write( "The RHS (right hand side) matches the original expression<BR>\n" );
document.write( "So,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 ALIGN=MIDDLE  ALT=\"%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3 ALIGN=MIDDLE  ALT=\"2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all real numbers x.\r<BR>\n" );
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document.write( "<font size=4 color=red>Therefore, the final answer is choice A</font>\r<BR>\n" );
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document.write( "Through graphing technology, we can use programs like <a  rel=nofollow HREF= \"https://www.geogebra.org/\">GeoGebra</a> to visually verify we have the proper answer.\r<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph below shows a red and blue striped curve. This is really a solid red curve underneath with a blue dashed curve directly on top of it. <BR>\n" );
document.write( "f(x) = red solid curve, g(x) = blue dashed curve<BR>\n" );
document.write( "The table of values show that f(x) = g(x) for any real x. While this isn't an exhaustive proof by any means, it's a quick way to verify the answer. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"60%\" src = \"https://i.imgur.com/aUdjlxg.png\"><BR>\n" );
document.write( "I recommend you play around with GeoGebra (or any equivalent graphing technology) to turn off/on the second graph to see how it is directly over top the first graph.<BR>\n" );
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