document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1140909>Question 1140909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The pistons in a car oscillate according to a sine function. The amplitude of the oscillation is 2, the period is 6 pi, and the phase shift is pi by 2 to the left. Now how would I write a formula to model the position? that is, the position of the piston, p, at time t seconds and how would I graph the equation?\r<BR>\n" );
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document.write( "6 Pi: https://i.imgur.com/aztBUqG.png\r<BR>\n" );
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document.write( "Pi by 2: https://i.imgur.com/UidKHjx.png\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #761593 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=761593\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your equation is y = 2 * sin ( 60 * ( x - 1.5 ) ) when you are working in degrees.\r<BR>\n" );
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document.write( "it looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2019/051501.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "i believe your equation is y = 2 * sin ( pi/3 *(x - 3/2 ) ) when you are working in radians.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/051502.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "you will notice that both graphs have a period of 6 and that both graphs are identical to each other.\r<BR>\n" );
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document.write( "y represents the position of the pistons which represents p for graphing purposes.\r<BR>\n" );
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document.write( "x represents time, which represents the variable t for graphing purposes.\r<BR>\n" );
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document.write( "in other words, the identical equations are:\r<BR>\n" );
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document.write( "p = 2 * sin ( 60 * ( t - 1.5 ) ) when you are working in degrees and .....\r<BR>\n" );
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document.write( "p = 2 * sin ( pi/3 *(t - 3/2 ) ) when you are working radians.\r<BR>\n" );
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document.write( "the biggest problem i had with this was the horizontal shift.\r<BR>\n" );
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document.write( "the first equation i worked was in degrees.\r<BR>\n" );
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document.write( "the phase shift to the left was given as pi / 2.\r<BR>\n" );
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document.write( "that's in radians.\r<BR>\n" );
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document.write( "i converted it to degrees by multiplying it by 180 / pi.\r<BR>\n" );
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document.write( "pi / 2 * 180 / pi  = 90 degrees.\r<BR>\n" );
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document.write( "that, i believe, is the phase shift if the periods was the normal period of 360 degrees.\r<BR>\n" );
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document.write( "i then needed to convert that phase shift to when the frequency was 60 rather than 1.\r<BR>\n" );
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document.write( "that made the phase shift equal to 90 / 60 = 1.5.\r<BR>\n" );
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document.write( "the amplitude was set at 2.\r<BR>\n" );
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document.write( "the period was set at 6.\r<BR>\n" );
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document.write( "the frequency was 360 / 6 = 60.\r<BR>\n" );
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document.write( "that led to the equation you see on the first graph.\r<BR>\n" );
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document.write( "when working in radians, ......\r<BR>\n" );
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document.write( "the amplitude was still 2.\r<BR>\n" );
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document.write( "the period was still 6.\r<BR>\n" );
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document.write( "the frequency became 2 * pi / 6 = 2/6 * pi = pi / 3.\r<BR>\n" );
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document.write( "the phase shift became (pi / 2) / (pi / 3) which became (pi / 2) * (3 / pi)  which became 3 / 2.\r<BR>\n" );
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document.write( "that's the same as 2.5, even though i left it at 3 / 2.\r<BR>\n" );
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document.write( "that led to the equation you see on the second graph.\r<BR>\n" );
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document.write( "i had a hell of a time finding references to phase shift in relationship to frequency, but i think i got a hint from the following reference.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathway.com/examples/trigonometry/graphing-trigonometric-functions/amplitude-period-and-phase-shift?id=342\" target = \"_blank\">https://www.mathway.com/examples/trigonometry/graphing-trigonometric-functions/amplitude-period-and-phase-shift?id=342</a>\r<BR>\n" );
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document.write( "what this is saying is, i think, .....\r<BR>\n" );
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document.write( "formula is y = sin ( 60 * x + 90) which is then translated to:\r<BR>\n" );
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document.write( "y = sin (60 * (x + 90 / 60) which then becomes y = sin (60 * (x + 1.5)\r<BR>\n" );
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document.write( "the shifting rules are:\r<BR>\n" );
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document.write( "if the shift is to the left, then it's x + c.\r<BR>\n" );
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document.write( "if the shift is to the right, then it's x - c.\r<BR>\n" );
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document.write( "this was probably the most confusing part for me and i hope that i got it right.\r<BR>\n" );
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document.write( "the degree formula just didn't work when i did it any other way.\r<BR>\n" );
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document.write( "this way, both the degree formula and the radian formula give the same answer which is a pretty good clue that it's probably right.\r<BR>\n" );
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