document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1116894>Question 1116894</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sinusoidal function has an amplitude of 3 units, a maximum at (0,4), and a period of 180 degrees. Represent the function using both sine and cosine functions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #731827 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=731827\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "We want equations in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Asin%28b%28x-c%29%29%2Bd ALIGN=MIDDLE  ALT=\"a%2Asin%28b%28x-c%29%29%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Acos%28b%28x-c%29%29%2Bd ALIGN=MIDDLE  ALT=\"a%2Acos%28b%28x-c%29%29%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The function has an amplitude of 3; so a in both equations is 3.<br><BR>\n" );
document.write( "The function has a maximum value of 4.  An amplitude of 3 and a maximum value of 4 means the minimum value is -2, and the centerline (d in each equation) 1.<br><BR>\n" );
document.write( "The period is 180 degrees, which is half the period of the basic sine or cosine function.  That means b is 2.<br><BR>\n" );
document.write( "Those are the relatively easy parts of the problem.  We have as the two equations<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%282%28x-c%29%29%2B1 ALIGN=MIDDLE  ALT=\"3%2Asin%282%28x-c%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Acos%282%28x-c%29%29%2B1 ALIGN=MIDDLE  ALT=\"3%2Acos%282%28x-c%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By far the hardest part (for most students) is finding the values for c in each equation.  That value determines the phase (horizontal) shift for the function.<br><BR>\n" );
document.write( "The given function has a maximum at x=0.  The basic cosine function has a maximum value at x=0, so there is no phase shift.  So the cosine equation for your example is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Acos%282x%29%2B1 ALIGN=MIDDLE  ALT=\"3%2Acos%282x%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For the sine function, the maximum occurs 1/4 of the way through the period; since the period of the function is 180 degrees (pi radians), the phase shift is 45 degrees (pi/4 radians) to the left.  So the phase shift c is -pi/4 (making \"x-c\" equal to \"x+pi/4\"), and the sine function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%282%28x%2Bpi%2F4%29%29%2B1 ALIGN=MIDDLE  ALT=\"3%2Asin%282%28x%2Bpi%2F4%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%282x%2Bpi%2F2%29%2B1 ALIGN=MIDDLE  ALT=\"3%2Asin%282x%2Bpi%2F2%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The first form is more meaningful to me, because I can see the phase shift of pi/4 clearly. </SPAN>\n" );
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