pre calc

in many parts of the coutry the average temp of a particular area may be modeled by a sin usoidal function. According to NCDC the average temp of k city is 92.2 in july and 32.2 in january assume that the period is 12 what is the best model for when july is month 0 and when january is month 0

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  1. first take Jan as month 0 (also month 12)
    then July is month 6 (note Dec is month 11)

    n angle in radians with period 12
    0Jan 0
    1feb 1*2pi/12 = pi/6
    2Mar 2*2 pi/12 = pi/3
    3Apr 3*2pi/12 = pi/2
    4may 4*2pi/12 = 2 pi/3
    5Jun 5*2pi/12 = 5pi/6
    6Jul 6*2 pi/12 = pi (half the period)
    7Aug 7*2 pi/12 = 7 pi/6
    8sept 8*2 pi/12 = 4 pi/3
    9oct 9*2 pi/12 = 3 pi/2
    10Nov 10*2pi/12 = 5pi/3
    11Dec 11*2pi/12 = 11pi/6
    12Jan 12*2pi/12 = 2 pi full circle
    note - angle is n pi/6
    so
    min at n = 0
    so
    T = a - b cos (n pi/6)
    32.2 = a - b cos 0 = a-b
    92.2 = a - b cos pi = a+b

    a-b = 32.2
    a+b = 92.2
    solve for a and b
    then
    T = a - b cos (n pi/6)

    go through that again with n = 0 at July

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