Modeling with Trigonometric Functions

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 91 degrees occurs at 5 PM and the average temperature for the day is 75 degrees. Find the temperature, to the nearest degree, at 6 AM.

Answer 1

Our model is T(t) = A sin[ω(t - t_o)] + B, and to get maximum T at 5pm = 17.00 hours on 24 hour clock, 91 = A sin(π/2) + 75; the amplitude A must be 91 - 75 = 16. The angular frequency ω has to be (2π radians/24 hours) = π/12. Now set ω(17 - t_o) = π/12 (17 - t_o) = π/2 resulting in t_o = 11.00 hours, which makes sense to be 6 hours ahead of the maximum of the sine curve at π/2 radians.

Now the model temperature at 6 am is T(6) = 16 sin[π/12 (6 - 11)] + 75 = 16 (-0.966) + 75 ≅ 60°. (sin[-5π/12] value is found by setting your calculator to RADians mode, not DEGrees mode).