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Math Help - Finding a Trig Equation

  1. #1
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    Finding a Trig Equation

    The desert temperature, H, oscillates daily between 39F at 6am and 79F at 6pm. Write a possible formula for H in terms of t, measured in hours from 6am. Use the form:

    H = A cos(Bt) + C.






    What I tried was this: I subtracted 39 from 79 to get 40, then divided that in half. That should be the amplitude A, right? Then I took 12 hours (from 6am to 6pm), and set that equal to 2pi / b, the period for a cosine function. I got pi/6, which should be B. Then to get the y axis up to 40, C should be 20 since the amplitude is 20.

    However, when I submitted this, all of them were wrong. Having tried everything I can think of, I'm asking for help. I'd appreciate any responses! Thank you!
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  2. #2
    Super Member Matt Westwood's Avatar
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    For a start C ought to be 59: if H goes from 39 to 79 then the point in the middle where cos Bt = 0 is when C = 59.

    Next, note that the time starts when H is at its lowest. Thus when t = 0, H = 59 - 20 cos (0) (note that minus sign). So A will have to be -20, not 20.

    Next note that your period is 24 not 12 - 12 hours is *half* the period.

    t = 0 means Bt = 0, t = 24 means Bt = 2 \pi so B = 2 \pi / 24 and so B = \pi / 12.

    Draw a graph - it may clarify things.
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  3. #3
    Senior Member pacman's Avatar
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    equation, H = A cos(Bt) + C.

    @ t = 0, H = 39F, then
    39 = A( cos 0) + C
    39 = A(1) + C
    39 = A + C ---- (1)
    -------------------------------------

    @ t = 12, H = 79F, then
    79 = A cos B(12) + C ---- (2)
    -------------------------------------

    39 = A + C ---- (1)
    79 = A cos B(12) + C ---- (2) - subtract (1) from (2)
    -------------------------------------

    40 = A(1 + cos 12 B) + 0;

    (40)(1) = (A)(1 + cos 12 B)

    equate factors, A = 40 and (1 + cos 12B) = 1.

    then, A = 40

    (1 + cos 12B) = 0

    cos 12B = -1

    12B = arccos(-1) = 180 degrees 0r pi rad

    then, B = pi/12

    period of the cosine function is 2pi/B = 2pi/(pi/12) = 24

    What is C? For H = 40 cos (pi/12)t + C

    is it 39F?
    Last edited by pacman; September 11th 2009 at 12:19 AM.
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  4. #4
    Senior Member pacman's Avatar
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    For H= 40cos ((pi/12)t) + 39 , for t = 0 to 24 hours . i will plot it below . . . . What is wrong with this?
    Attached Thumbnails Attached Thumbnails Finding a Trig Equation-pi10.gif  
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  5. #5
    Senior Member pacman's Avatar
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    For H = 40 cos (pi/12)t + C, for t = - 6 hours, cos pi/2 = 0, H = C = 39.

    Since t = -6 hours corresponds to 12am, and pi/2 was added tp compensate for the 12 hours lag, then the new equation will be

    H = 40 cos ((pi/12)t - pi/2) + 39

    see my my graph.

    i am still reluctant to post this, i believe there is something wrong with it. HELP

    the 6pm H does not corresponds to the given value
    Attached Thumbnails Attached Thumbnails Finding a Trig Equation-pi-11.gif  
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  6. #6
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by pacman View Post
    equation, H = A cos(Bt) + C.

    @ t = 0, H = 39F, then
    39 = A( cos 0) + C
    39 = A(1) + C
    39 = A + C ---- (1)
    -------------------------------------

    @ t = 12, H = 79F, then
    79 = A cos B(12) + C ---- (2)
    -------------------------------------

    39 = A + C ---- (1)
    79 = A cos B(12) + C ---- (2) - subtract (1) from (2)
    -------------------------------------

    etc.
    I think this is where you went wrong.(2) - (1) gives:
    40 = A(cos 12 B - 1)
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  7. #7
    Senior Member pacman's Avatar
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    Thanks Matt Westwood,
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  8. #8
    Senior Member pacman's Avatar
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    This is the plot of Matt Westwood's equation, H = 69 - 20cos (pi/12)t, for t = 0 to 24 hours.

    I believe this is the right solution
    Attached Thumbnails Attached Thumbnails Finding a Trig Equation-pi-1.gif  
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  9. #9
    Senior Member pacman's Avatar
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    the above post is still wrong, H = 59 - 20 cos (pi/12)t, for t = 0 to 24 hours
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  10. #10
    Senior Member pacman's Avatar
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    this is the graph of H = 59 - 20 cos (pi/12)t, for t = 0 to 24 hours
    Attached Thumbnails Attached Thumbnails Finding a Trig Equation-pi-3.gif  
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  11. #11
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by pacman View Post
    This is the plot of Matt Westwood's equation, H = 69 - 20cos (pi/12)t, for t = 0 to 24 hours.

    I believe this is the right solution
    59 not 69, yeah?
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  12. #12
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by pacman View Post
    this is the graph of H = 59 - 20 cos (pi/12)t, for t = 0 to 24 hours
    What I said.
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