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The number of daylight hours in a day is harmonic. Suppose that in a particular location, the shortest day of the year has 7 hours of daylight and the longest day of the year has 18 hours. Then, we can model its motion with the function N=Asin(Bt) + C where t is expressed in days and A, B > 0. Find A and B, using 365 days for one year.

I need to find what A, B, and C equal. Thanks.

  • Trigonometry -

    In the equation:
    N=Asin(Bt) + C
    A represents half of the difference between the maximum and minimum, sometimes called amplitude.
    B is a multiplitive factor to ensure that the cycle of sine (2π) fits into the physical cycle, in this problem, 365 days.
    Since the amplitude of daylight hours is (18-7)/2=5.5, we obtain A=5.5.
    We need a value of B such that when t=0 and t=365, Bt becomes 0 and 2π.
    So 365B = 2π, or B=2π/365.

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