document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202255>Question 1202255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi can you please help me with these sinusoidal functions questions? Please include all answers in degree mode and not in radian. Thank you!\r<BR>\n" );
document.write( "\n" );
document.write( "1. State the transformations that have been applied to y = cos x to obtain the function y = 1.78cos(x - 120°)+ 10\r<BR>\n" );
document.write( "\n" );
document.write( "2. \r<BR>\n" );
document.write( "\n" );
document.write( "3. \r<BR>\n" );
document.write( "\n" );
document.write( "4. Determine the equation of a sine function that would have a range of {y∈R | -4 ≤ y ≤ 1} and a period of 45°. Determine the cosine function that results in the same graph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836957 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=836957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll focus on problem 4 to get you started.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The template for the sine function is <BR>\n" );
document.write( "y = A*sin(B(x-C))+D\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "|A| = amplitude<BR>\n" );
document.write( "B = used to find the period<BR>\n" );
document.write( "C = phase shift<BR>\n" );
document.write( "D = midline\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The next four sections will go over how to find the values for A,B,C,D\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Determine the value of A.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Range is {y∈R | -4 ≤ y ≤ 1}<BR>\n" );
document.write( "y = -4 = min<BR>\n" );
document.write( "y = 1 = max<BR>\n" );
document.write( "The vertical distance between these y values is 5 units. <BR>\n" );
document.write( "Count the spaces, or subtract and use absolute value, to determine this distance. A vertical number line might help visualize things.<BR>\n" );
document.write( "Half of this vertical distance is 5/2 = 2.5<BR>\n" );
document.write( "We either can have A = 2.5 or A = -2.5<BR>\n" );
document.write( "I'll go with A = 2.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Determine the value of B.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We want a period of 45 degrees<BR>\n" );
document.write( "T = period = 45<BR>\n" );
document.write( "B = 360/T<BR>\n" );
document.write( "B = 360/45<BR>\n" );
document.write( "B = 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Determine the value of C.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The phase shift can be whatever we want. Let's go for C = 0 as it's easiest.<BR>\n" );
document.write( "There's not much else to this section.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Determine the value of D.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Range is {y∈R | -4 ≤ y ≤ 1}<BR>\n" );
document.write( "y = -4 = min<BR>\n" );
document.write( "y = 1 = max<BR>\n" );
document.write( "Apply the midpoint formula to get (-4+1)/2 = -3/2 = -1.5<BR>\n" );
document.write( "This is the value of D\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Summary<BR>\n" );
document.write( "A = 2.5<BR>\n" );
document.write( "B = 8<BR>\n" );
document.write( "C = 0<BR>\n" );
document.write( "D = -1.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We therefore go from<BR>\n" );
document.write( "y = A*sin(B(x-C))+D<BR>\n" );
document.write( "to<BR>\n" );
document.write( "y = 2.5*sin(8(x-0))+(-1.5)<BR>\n" );
document.write( "and it fully simplifies to<BR>\n" );
document.write( "<font color=red>y = 2.5*sin(8x)-1.5</font><BR>\n" );
document.write( "That is one possible sine equation that has a period of 45 degrees and has a range of  -4 ≤ y ≤ 1 where y is a real number<BR>\n" );
document.write( "I recommend using Desmos to graph out the function to verify the answer\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/9qhl97rerg\">https://www.desmos.com/calculator/9qhl97rerg</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "GeoGebra is another useful graphing app.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The name \"cosine\" heavily implies that we have a sinusoidal curve. <BR>\n" );
document.write( "\"sine\" is buried in the name \"cosine\". <BR>\n" );
document.write( "Cosine is a phase-shifted version of sine.<BR>\n" );
document.write( "For more information, search out \"trig co-functions\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To go from sine to cosine, we involve a phase shift of 90 degrees<BR>\n" );
document.write( "This is due to these two identities <BR>\n" );
document.write( "sin(x) = cos(90-x)<BR>\n" );
document.write( "cos(x) = sin(90-x)<BR>\n" );
document.write( "Why do these identities work? Draw out a right triangle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-4%2C4%2C-4%2C4%2C%0D%0Aline%28-3%2C-3%2C3%2C-3%29%2C%0D%0Aline%283%2C-3%2C3%2C3%29%2C%0D%0Aline%283%2C3%2C-3%2C-3%29%2C%0D%0A%0D%0Aline%283-0.2%2C-3%2C3-0.2%2C-3%2B0.2%29%2C%0D%0Aline%283-0.2%2C-3%2B0.2%2C3%2C-3%2B0.2%29%2C%0D%0A%0D%0Alocate%28-3%2B0.5%2C-3%2B0.5%2C%22x%22%29%2C%0D%0Alocate%283-0.8%2C3-0.8%2C%2290-x%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point, it should be fairly clear why those identities work.<BR>\n" );
document.write( "Let me know if you are stuck on this part.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we'll go from<BR>\n" );
document.write( "y = 2.5*sin(8x)-1.5<BR>\n" );
document.write( "to<BR>\n" );
document.write( "<font color=\"red\">y = 2.5*cos(90-8x)-1.5</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/lobh6fe5te\">https://www.desmos.com/calculator/lobh6fe5te</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );