document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183523>Question 1183523</A>: <I><SPAN STYLE=\"word-break: break-all;\"> rewrite: y=2sin(πt) - 3cos(πt) in y=Asin(Bt + C) form, using sum formula </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813896 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=813896\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm going to use this identity<BR>\n" );
document.write( "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)<BR>\n" );
document.write( "to say the following:<BR>\n" );
document.write( "2*sin(pi*t) - 3*cos(pi*t) = A*sin(Bt + C)<BR>\n" );
document.write( "2*sin(pi*t) - 3*cos(pi*t) = A*(sin(Bt)*cos(C) + cos(Bt)*sin(C))<BR>\n" );
document.write( "2*sin(pi*t) - 3*cos(pi*t) = A*sin(Bt)*cos(C) + A*cos(Bt)*sin(C)<BR>\n" );
document.write( "2*sin(pi*t) - 3*cos(pi*t) = A*cos(C)*sin(Bt) + A*sin(C)*cos(Bt)\r<BR>\n" );
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document.write( "Let's highlight a pair of matching terms on either side. I'll use <font color=red>red</font> to do so<BR>\n" );
document.write( "<font color=red>2*sin(pi*t)</font> - 3*cos(pi*t) = <font color=red>A*cos(C)*sin(Bt)</font> + A*sin(C)*cos(Bt)\r<BR>\n" );
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document.write( "Based on those highlighted items, we know that<BR>\n" );
document.write( "2*sin(pi*t) = A*cos(C)*sin(Bt)<BR>\n" );
document.write( "which must lead to <BR>\n" );
document.write( "A*cos(C) = 2<BR>\n" );
document.write( "B = pi\r<BR>\n" );
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document.write( "The non-highlighted items on either side of that equation lead us to<BR>\n" );
document.write( "-3*cos(pi*t) = A*sin(C)*cos(Bt)<BR>\n" );
document.write( "which leads to<BR>\n" );
document.write( "A*sin(C) = -3\r<BR>\n" );
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document.write( "To summarize:<BR>\n" );
document.write( "A*sin(C) = -3<BR>\n" );
document.write( "A*cos(C) = 2\r<BR>\n" );
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document.write( "Squaring both sides for each equation yields<BR>\n" );
document.write( "A^2*sin^2(C) = 9<BR>\n" );
document.write( "A^2*cos^2(C) = 4\r<BR>\n" );
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document.write( "Add up those equations and solve for A<BR>\n" );
document.write( "A^2*sin^2(C)+A^2*cos^2(C) = 9+4<BR>\n" );
document.write( "A^2*(sin^2(C)+cos^2(C)) = 13<BR>\n" );
document.write( "A^2*1 = 13<BR>\n" );
document.write( "A^2 = 13<BR>\n" );
document.write( "A = sqrt(13)\r<BR>\n" );
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document.write( "We can then update the equations<BR>\n" );
document.write( "A*sin(C) = -3<BR>\n" );
document.write( "A*cos(C) = 2<BR>\n" );
document.write( "into this<BR>\n" );
document.write( "sqrt(13)*sin(C) = -3<BR>\n" );
document.write( "sqrt(13)*cos(C) = 2\r<BR>\n" );
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document.write( "Pick any of those latter two equations to solve for C.<BR>\n" );
document.write( "sqrt(13)*sin(C) = -3<BR>\n" );
document.write( "sin(C) = -3/sqrt(13)<BR>\n" );
document.write( "C = arcsin(-3/sqrt(13))<BR>\n" );
document.write( "C = -0.98279372324732<BR>\n" );
document.write( "The value is approximate. Your calculator needs to be in radian mode. \r<BR>\n" );
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document.write( "Therefore, we have<BR>\n" );
document.write( "y = 2*sin(pi*t) - 3*cos(pi*t)<BR>\n" );
document.write( "turn into<BR>\n" );
document.write( "y = sqrt(13)*sin(pi*t - 0.9827937232473)<BR>\n" );
document.write( "which is in the form y = A*sin(Bt + C)\r<BR>\n" );
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document.write( "Final Answer:<BR>\n" );
document.write( "y = sqrt(13)*sin(pi*t - 0.9827937232473)<BR>\n" );
document.write( "This equation is approximate \r<BR>\n" );
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document.write( "</font> </SPAN>\n" );
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