document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=889554>Question 889554</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value of A when cos 2A=sin 3A. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #538387 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=538387\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i was able to find the solution graphically.<BR>\n" );
document.write( "that solution is shown below in the following 4 pictures.<BR>\n" );
document.write( "first picture takes you from -400 degrees to 0 degrees.<BR>\n" );
document.write( "second picture takes you from 0 degrees to 400 degrees.<BR>\n" );
document.write( "third picture takes you from 400 degrees to 800 degrees.<BR>\n" );
document.write( "fourth picture takes you from 800 degrees to 1200 degrees.<BR>\n" );
document.write( "those graphs are enough to show the periodicity of the solutions.<BR>\n" );
document.write( "those solutions and their periodicity are shown here.<BR>\n" );
document.write( "x = 18 degrees +/- 360 degrees.<BR>\n" );
document.write( "x = 162 degrees +/- 360 degrees.<BR>\n" );
document.write( "x = 90 degrees plus or minus 360 degrees.<BR>\n" );
document.write( "x = 234 degrees +/- 360 degrees.<BR>\n" );
document.write( "x = 306 degrees +/- 360 degrees.<BR>\n" );
document.write( "that should be all the possible solutions of the equation cos(2x) = sin(3x).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/072801.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/072802.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/072803.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/072804.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "if you take any one of those solutions and add or subtract 360 from it, you should get the other solutions shown.<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "18 - 360 = -342<BR>\n" );
document.write( "162 - 360 = -198<BR>\n" );
document.write( "90 - 360 = -270<BR>\n" );
document.write( "234 - 360 = -126<BR>\n" );
document.write( "306 - 360 = -54<BR>\n" );
document.write( "look on the first graph and you should see all those solutions.<BR>\n" );
document.write( "going the other way, for one period up, you get:<BR>\n" );
document.write( "18 + 360 = 378<BR>\n" );
document.write( "162 + 360 = 522<BR>\n" );
document.write( "90 + 360 = 450<BR>\n" );
document.write( "234 + 360 = 594<BR>\n" );
document.write( "306 + 360 = 666<BR>\n" );
document.write( "look on the second graph and you should see 378.<BR>\n" );
document.write( "look on the third graph and you should see the rest of those solutions.\r<BR>\n" );
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document.write( "i also found an algebraic solution on the web at the following link:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://mathcentral.uregina.ca/QQ/database/QQ.09.04/pierre1.html\" target = \"_blank\">http://mathcentral.uregina.ca/QQ/database/QQ.09.04/pierre1.html</a>\r<BR>\n" );
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document.write( "that solution looked good except for the fact that it appeared to be missing the solution of cos(2x) = sin(3x) = -1\r<BR>\n" );
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document.write( "i then decided to use their strategy and solve the equation for myself to see what was missing and why.\r<BR>\n" );
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document.write( "my algebraic solution is shown below.\r<BR>\n" );
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document.write( "start with cos(2A) = sin(3A)\r<BR>\n" );
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document.write( "cos(2A) is equivalent to cos^2(A) - sin^2(A)\r<BR>\n" );
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document.write( "sin(3A) is equivalent to sin(A + 2A)\r<BR>\n" );
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document.write( "sin(A + 2A) is equivalent to sin(A)*cos(2A) + cos(A)*sin(2A)\r<BR>\n" );
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document.write( "sin(2A) is equivalent to 2*sin(A)*cos(A)\r<BR>\n" );
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document.write( "sin(A + 2A) is therefore equal to sin(A)*(cos^2(A)-sin^2(A)) + cos(A)*2*sin(A)*cos(A)\r<BR>\n" );
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document.write( "replacing the original expressions with their equivalent expressions, we get:\r<BR>\n" );
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document.write( "start with cos(2A) = sin(3A) becomes:\r<BR>\n" );
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document.write( "cos^2(A) - sin^2(A) = sin(A)*(cos^2(A)-sin^2(A)) + cos(A)*2*sin(A)*cos(A)\r<BR>\n" );
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document.write( "we can distribute the multiplication and combine like terms to get:\r<BR>\n" );
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document.write( "cos^2(A) - sin^2(A) = sin(A)*cos^2(A) - sin^3(A) + 2*sin(A)*cos^2(A)\r<BR>\n" );
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document.write( "we can combine like terms again to get:\r<BR>\n" );
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document.write( "cos^2(A) - sin^2(A) = 3*sin(A)*cos^2(A) - sin^3(A)\r<BR>\n" );
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document.write( "if we add sin^3(A) to both sides of the equation and we subtract 3*sin(A)*cos^2(A) from both sides of the equation, we get:\r<BR>\n" );
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document.write( "cos^2(A) - sin^2(A) - 3*sin(A)*cos^2(A) + sin^3(A) = 0\r<BR>\n" );
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document.write( "if we replace cos^2(A) with its identical expression of 1 - sin^2(A), then we get:\r<BR>\n" );
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document.write( "1 - sin^2(A) - sin^2(A) - 3*sin(A)*(1-sin^2(A)) + sin^3(A) = 0\r<BR>\n" );
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document.write( "if we combine like terms and distribute the multiplication, we get:\r<BR>\n" );
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document.write( "1 - 2*sin^2(A) - 3*sin(A) + 3*sin^3(A) + sin^3(A) = 0\r<BR>\n" );
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document.write( "if we combine like terms again, we get:\r<BR>\n" );
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document.write( "1 - 2*sin^2(A) - 3*sin(A) + 4*sin^3(A) = 0\r<BR>\n" );
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document.write( "if we re-arrange the terms in descending order of degrees, we get:\r<BR>\n" );
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document.write( "4*sin^3(A) - 2*sin^2(A) - 3*sin(A) + 1 = 0\r<BR>\n" );
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document.write( "one of the factors of this equation is sin(A) - 1, which we found through synthetic division.\r<BR>\n" );
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document.write( "the equation becomes:\r<BR>\n" );
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document.write( "(sin(A)-1) * (4sin^2(A) + 2sin(A) - 1) = 0\r<BR>\n" );
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document.write( "one of our solutions will be sin(A) = 1\r<BR>\n" );
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document.write( "through the use of the quadratic formula, our other solutions will be:\r<BR>\n" );
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document.write( "sin(A) = .3090169944 or sin(A) = .8090169944\r<BR>\n" );
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document.write( "solving for A, we get:\r<BR>\n" );
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document.write( "A = 18 degrees or A = -54 degrees.\r<BR>\n" );
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document.write( "those are our solutions for one period only.<BR>\n" );
document.write( "the graph shows us the other solutions from which we could include the periodicity to these solutions.\r<BR>\n" );
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document.write( "the solutions that we solved for algebraically are:\r<BR>\n" );
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document.write( "18 degrees<BR>\n" );
document.write( "-54 degrees<BR>\n" );
document.write( "90 degrees\r<BR>\n" );
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document.write( "when you use these solutions.....\r<BR>\n" );
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document.write( "cos(2*18) = cos(36) = .809.....<BR>\n" );
document.write( "sin(3*18) = sin(54) = .809...\r<BR>\n" );
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document.write( "cos(2*-54) = cos(-108) = -.309...<BR>\n" );
document.write( "sin(3*-54) = sin(-162) = -.309...\r<BR>\n" );
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document.write( "cos(2*90) = cos(180) = -1<BR>\n" );
document.write( "sin(3*90) = sin(270) = -1\r<BR>\n" );
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document.write( "the graphical solution and the algebraic solution now match for all values shown in the algebraic solution so i'm satisfied that the algebraic solution was solved successfully.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );