document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=866819>Question 866819</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, any help with the following question would be greatly appreciated. Using an analytical method (compound angle formula) solve the following trigonometric equation for angles in the range: sin&#952;=3sin(&#952;-Pi/6). Substitute your answers into the original expression to check that they are valid. <BR>\n" );
document.write( "Thanks in advance.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #522535 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=522535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'll use x instead of theta.<BR>\n" );
document.write( "doesn't change the problem.<BR>\n" );
document.write( "it's just easier to type.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "sin(x) = 3sin(x-pi/6)\r<BR>\n" );
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document.write( "since sin(d-e) = sin(d)*cos(e) - cos(d)*sin(e), then if d = x and e = pi/6, you get:\r<BR>\n" );
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document.write( "sin(x) = 3 * [sin(x)*cos(pi/6) -  cos(x)*sin(pi/6)]\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "sin(x) = 3*sin(x)*cos(pi/6) - 3*cos(x)*sin(pi/6)\r<BR>\n" );
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document.write( "you can re-write this to be:\r<BR>\n" );
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document.write( "sin(x) = (3*cos(pi/6))*sin(x) - (3*sin(pi/6))*cos(x)\r<BR>\n" );
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document.write( "if you let a = 3*cos(pi/6) and you let b = 3*sin(pi/6), then you get:\r<BR>\n" );
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document.write( "sin(x) = a*sin(x) - b*cos(x)\r<BR>\n" );
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document.write( "subtract sin(x) from both sides of this equation and add b*cos(x) to both sides of this equation to get:<BR>\n" );
document.write( "b*cos(x) = a*sin(x) - sin(x)<BR>\n" );
document.write( "factor out the sin(x) in this equation to get:<BR>\n" );
document.write( "b*cos(x) = sin(x) * (a-1)<BR>\n" );
document.write( "divide both sides of this equation by b and divide both sides of this equation by sin(x) to get:<BR>\n" );
document.write( "cos(x) / sin(x) = (a-1) / b<BR>\n" );
document.write( "since cot(x) = cos(x) / sin(x),  this equation becomes:<BR>\n" );
document.write( "cot(x) = (a-1) / b\r<BR>\n" );
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document.write( "the equation you are working with now is:\r<BR>\n" );
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document.write( "cot(x) = (a-1) / b\r<BR>\n" );
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document.write( "now you want to find the value of a and b and substitute those values for a and b in the equation.\r<BR>\n" );
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document.write( "a = 3 * cos(pi/6) which makes a = 2.598076211<BR>\n" );
document.write( "b = 3 * sin(pi/6) which makes b = 1.5\r<BR>\n" );
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document.write( "replace a and b in the formula of cot(x) = (a-1) / b to get:\r<BR>\n" );
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document.write( "cot(x) = (2.598076211 - 1) / 1.5<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "cot(x) = 1.598076211 / 1.5<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "cot(x) = 1.065384141\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = arc-cot(1.065384141) = .753751599\r<BR>\n" );
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document.write( "that should be your answer.\r<BR>\n" );
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document.write( "in the original equation, replace x with .753751599 to get:\r<BR>\n" );
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document.write( "sin(x) = 3sin(x-pi/6) becomes:\r<BR>\n" );
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document.write( "sin(.753751599) = 3sin(.753751599-pi/6) which becomes:\r<BR>\n" );
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document.write( ".684378960 = .684370960\r<BR>\n" );
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document.write( "you did good.\r<BR>\n" );
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document.write( "don't forget to set your calculator to radians in order to solve this problem.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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