document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=75884>Question 75884</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Please help me<BR>\n" );
document.write( "tan(3x)+1={square root(2)}sec(3x)\r<BR>\n" );
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document.write( "so far i have tried:<BR>\n" );
document.write( "(tan(3x)+1)^2=(sq rt(2))^2(sec(3x))^2<BR>\n" );
document.write( "tan^2(9x^2)+1^2=2sec^2(9x^2)<BR>\n" );
document.write( "tan^2 +1=2sec^2<BR>\n" );
document.write( "tan^2+1 = sec^2<BR>\n" );
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document.write( "   2\r<BR>\n" );
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document.write( "I am totally confused now.  This problem was assigned by my teacher.  <BR>\n" );
document.write( "We are using Trigonometry a Unit Circle Approach  7th edition<BR>\n" );
document.write( "by Sullivan  ISBN # 0-13-143111-0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #54569 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><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=54569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "Ok, the trick with these things until your comfortable with them is to get rid of the trigonometry as soon as possible and go back to good old fashioned algebra. To get rid of the trigonometry the first thing to do is write the equation in terms of only one trig function. I will choose <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\tan(3x)\">, so we're trying to replace the secant with a tangent.\r<BR>\n" );
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document.write( "Obvioulsy we're after an identity connecting tangent and secant. How about this one?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?1+\tan^2\theta = \sec^2\theta\">\r<BR>\n" );
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document.write( "You may ask, but in my equation I have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\tan(3x)\"> This is easily solved by saying <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\theta=3x\">\r<BR>\n" );
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document.write( "So, hopefully you can rearrange the identity to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sec\(3x) = \pm\sqrt{1+\tan^2(3x)}\">\r<BR>\n" );
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document.write( "Putting this into our equation gives\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\tan(3x)+1=\pm\sqrt{2}\sqrt{1+\tan^2(3x)}\">\r<BR>\n" );
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document.write( "Now let's get rid of that horrible tangent by saying <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p=\tan(3x)\">\r<BR>\n" );
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document.write( "This leaves us with the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p+1=\pm\sqrt{2}\sqrt{1+p^2}\">\r<BR>\n" );
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document.write( "We will now square both side, but remember that when we square the equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?a=b\">, we will also have the solutions to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?a=-b\"> so you must check any solutions you get work in your original equation.\r<BR>\n" );
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document.write( "Anyway, squaring gives\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p^2+2p+1=2+2p^2\">\r<BR>\n" );
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document.write( "This rearanges nicely to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p^2-2p+1=0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?(p-1)^2=0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p=1\">\r<BR>\n" );
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document.write( "But we have solved for p, and we need to solve for x. Up above we defined\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p=\tan(3x)\">\r<BR>\n" );
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document.write( "But, remember that tangent is a periodic function, and I can add or subtract integer multiples of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?2\pi\"> (we're working in radians right?) so the actual equation I want to solve is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\tan(3x-2n\pi)=p=1, \forall n\in\mathbb{Z}\">\r<BR>\n" );
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document.write( "Taking inverse tangent and tidying up, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x=\frac{\pi}{12}(1+8n)\">\r<BR>\n" );
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document.write( "Leting n=0,1,2 gives the solutions <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{\pi}{12},\frac{9\pi}{12},\frac{17\pi}{12}\"> Don't forget to put these back in to your equation because when we squared we may have inadvertently considered solutions that don't solve our original equation. Question: if we didn't, can you say why not?\r<BR>\n" );
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document.write( "Hope some of that was helpful,<BR>\n" );
document.write( "Kev </SPAN>\n" );
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