document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1013983>Question 1013983</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 12. A sound wave is modeled with the equation y = 1/4 cos 2pi/3 &#952;. <BR>\n" );
document.write( "a. Find the period. Explain your method. <BR>\n" );
document.write( "b. Find the amplitude. Explain your method.\r<BR>\n" );
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document.write( "I need help with this! Thank you...  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #630295 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=630295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the basic formula for cosine is:\r<BR>\n" );
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document.write( "y = a * (b * (x-c)) + d\r<BR>\n" );
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document.write( "a is the amplitude.<BR>\n" );
document.write( "b is the frequency.<BR>\n" );
document.write( "c is the horizontal displacement.<BR>\n" );
document.write( "d is the vertical displacement.\r<BR>\n" );
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document.write( "your equation is y = 1/4 * cos (2*pi/3 * x).\r<BR>\n" );
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document.write( "i used x instead of theta.<BR>\n" );
document.write( "no matter.<BR>\n" );
document.write( "just replace my x with your theta and you have your answer.\r<BR>\n" );
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document.write( "the amplitude is 1/4.\r<BR>\n" );
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document.write( "the frequency is 2 * pi/3.\r<BR>\n" );
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document.write( "the horizontal displacement is 0.\r<BR>\n" );
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document.write( "the vertical displacement is 0.\r<BR>\n" );
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document.write( "if you are dealing in degrees, then the period is equal to 360 divided by the frequency.\r<BR>\n" );
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document.write( "if you are dealing in radians, then the period is equal to 2*pi divided by the frequency.\r<BR>\n" );
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document.write( "if you know the frequency, you can solve for the period using this formula.<BR>\n" );
document.write( "if you know the period, you can solve for the frequency using this formula.\r<BR>\n" );
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document.write( "consider f = frequency and p = period\r<BR>\n" );
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document.write( "in degrees:\r<BR>\n" );
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document.write( "p = 360 / f<BR>\n" );
document.write( "solve for f to get f = 360 / p\r<BR>\n" );
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document.write( "in radians:\r<BR>\n" );
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document.write( "p = 2pi / f<BR>\n" );
document.write( "solve for f to get f = 2pi / p\r<BR>\n" );
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document.write( "since you're in radians, we'll use the radian formula to get:\r<BR>\n" );
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document.write( "p = 2pi/(2pi/3)<BR>\n" );
document.write( "this is the same as p = 2pi * 3 / 2pi\r<BR>\n" );
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document.write( "the result is that p = 3.\r<BR>\n" );
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document.write( "if you graph this equation, your x-axis will be in radians and one full cycle of the cosine wave will take up 3 radians.\r<BR>\n" );
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document.write( "here's the graph of your equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/011502.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the normal period is 2pi in radians.\r<BR>\n" );
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document.write( "the equation of y = cos(x) is really y = cos(1*x).\r<BR>\n" );
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document.write( "1 is the frequency.\r<BR>\n" );
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document.write( "p = 2pi/f becomes p = 2pi/1 which becomes p = 2pi.\r<BR>\n" );
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document.write( "when the frequency is 1, the period is 2pi.\r<BR>\n" );
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document.write( "the following graph of y = cos(x) shows that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/011503.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "in both cases, the cosine was starts at 1 and ends at 1.\r<BR>\n" );
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document.write( "that's because cos(0) = 1 and cos(360) = 1\r<BR>\n" );
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document.write( "in radians, cos(0) = 1 and cos(2pi) = 1\r<BR>\n" );
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document.write( "also, cos(180) = -1 and cos(pi) = -1\r<BR>\n" );
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document.write( "when the period is 2pi, halfway through the period is pi.\r<BR>\n" );
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document.write( "when the period is 3, halfway through the period is 1.5.\r<BR>\n" );
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document.write( "convert 3 radians to degrees and you get 3 radians * 180 / pi = 171.8873385 degrees.\r<BR>\n" );
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document.write( "half of that is equal to 75.84355826 degrees.\r<BR>\n" );
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document.write( "convert that to radians and you get 75.84355826 degrees * pi / 180 = .4774648293 * pi radians which is equal to 1.5 radians.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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