document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169739>Question 1169739</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I don't know what the topic is \r<BR>\n" );
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document.write( "Background info:<BR>\n" );
document.write( "You are given the equation y(t)=2sin(4πt) + 5cos(4πt), which models the position of the weight, with respect to time. You need to find the amplitude of the oscillation, the angular frequency, and the initial conditions of the motion. You will also be required to find the time(s) at which the weight is at a particular position. To find this information, you need to convert the equation to the first form,y(t)=A sin(wt+Φ)<BR>\n" );
document.write( "More background info:<BR>\n" );
document.write( "y(t) = distance of weight from equilibrium position<BR>\n" );
document.write( "w= angular frequency (measured in radians per second)<BR>\n" );
document.write( "A = amplitude<BR>\n" );
document.write( "Φ = phase (depends on initial conditions)<BR>\n" );
document.write( "c1 = AsinΦ<BR>\n" );
document.write( "c2 = AcosΦ\r<BR>\n" );
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document.write( "Question: To rewrite 2sin(4πt) + 5cos(4πt) in the form y(t)=A sin(wt+ Φ), you must first find the amplitude, A. Use the given values c1=AsinΦ and c2=AcosΦ, along with the Pythagorean identity, to solve for A. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794605 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=794605\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y(t)=2sin(4πt) + 5cos(4πt)<BR>\n" );
document.write( "In order to rewrite the equation in the form y(t)=A sin(wt+ Φ), we use the <BR>\n" );
document.write( "identity y(t) = AcosΦsin(wt) + AsinΦcos(wt) <BR>\n" );
document.write( "Thus c1 = AcosΦ = 2, c2 = AsinΦ = 5<BR>\n" );
document.write( "If we divide the 1st by the 2nd, we have:<BR>\n" );
document.write( "tanΦ = 5/2 = 2.5 -> Φ = 1.1903<BR>\n" );
document.write( "Therefore, A = 5/sinΦ = 5.3852<BR>\n" );
document.write( "So the equation of motion is y(t) = 5.3852*sin(wt + 1.1903)<BR>\n" );
document.write( "The angular frequency, w = 4π<BR>\n" );
document.write( "The period, T = 2π/w = 1/2, and the initial condition is y(0) = 5/sinΦ*sin(Φ) = 5.<BR>\n" );
document.write( "One can also use the Pythagorean identity, as the problem suggests, to get the<BR>\n" );
document.write( "amplitude: c1^2 + c2^2 = 25 + 4 = A^2(sin^2Φ + cos^2Φ) -> A = sqrt(29) </SPAN>\n" );
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