The temperature during the day can be modeled by a sinusoid. Answer the following question given that the low temperature of 52 degrees occurs at 6 AM and the high temperature for the day is 84 degrees. Assuming $t$ is the number of hours since midnight, find an equation for the temperature, $T$, in terms of $t$.

Step 1 ：The temperature can be modeled by a sinusoidal function of the form \(T(t) = A \sin(B(t - C)) + D\), where \(A\) is the amplitude, \(B\) is the frequency, \(C\) is the phase shift, and \(D\) is the vertical shift.

Step 2 ：The amplitude \(A\) is half the difference between the high and low temperatures, which is \(A = \frac{84 - 52}{2} = 16.0\).

Step 3 ：The vertical shift \(D\) is the average of the high and low temperatures, which is \(D = \frac{84 + 52}{2} = 68.0\).

Step 4 ：The phase shift \(C\) is the time at which the minimum temperature occurs. Since the minimum temperature occurs at 6 AM, \(C = 6\).

Step 5 ：The period of the sinusoidal function is 24 hours because the temperature pattern repeats every 24 hours. Therefore, the frequency \(B\) is the reciprocal of the period, which is \(B = \frac{2\pi}{24} = \frac{\pi}{12}\).

Step 6 ：Substituting the values of \(A\), \(B\), \(C\), and \(D\) into the equation, we get \(T(t) = 16.0 \sin\left(\frac{\pi}{12}(t - 6)\right) + 68.0\).

Step 7 ：\(\boxed{T(t) = 16.0 \sin\left(\frac{\pi}{12}(t - 6)\right) + 68.0}\) is the equation for the temperature, \(T\), in terms of \(t\).