how to use trig function

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 70 degrees occurs at 4 PM and the average temperature for the day is 60 degrees. Find the temperature, to the nearest degree, at 5 AM.

Answer 1

The function can be expressed as y = Asin(Bx+C) + D, where x = number of hours past midnight.

 

Since the high for the day is 70° and the average is 60°, the amplitude = 70-60 = 10.

 

Period = 24.  So, 2π/B = 24.  B = π/12

 

D = average temp = 60

 

When x = 16, y = 70.  So, 70 = 10sin(4π/3+C) + 60

 

                                       sin(4π/3+C) = 1

 

                                             4π/3+C = π/2

 

                                                      C = -5π/6

 

So, y = 10sin(π/12x - 5π/6) + 60 

 

Temperature at 5 AM = 10sin(5π/12 - 5π/6) + 60

 

                               = 10sin(-5π/12) + 60 ≈ 50°