Trigonometry, modeling with sinusoidal functions question

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 38 and 72 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 51 degrees?

Answer 1

f(t) = Asin[B(t+C)+D = the temperature at time t

A=amplitude = (72-38)/2 = 34/2 =17

B= 2pi/period, period = 24 hours, B=2pi/24 = pi/12

C= phase shift=-8

f(t)= 55 = 17sin[pi/12(8+C)]+55

sin[pi/12(8+C)] = 0

pi/12(8+C) = 0

8+C=0

C=-8

D= midline = 38+17=72-17 = 55= average temperature which 1st occurs at 8 am

then again at 8pm, 72 high at 2pm, low 38 at 2am

f(t) = 17sin[pi/12(t-8)]+55 = 51

solve for t