We are told the temperature can be modeled as a sinusoidal function. This means the temperature (T) at a given time, t, can be modelled as T(t)=sin(ωt + Φ) + C, where ω, Φ, and C are constants that we must find to make the model fit the data. Specifically, ω is what we call the "angular frequency" and Φ is what we call the "phase shift". The constant ω symbolically represents how fast the system oscillates. A pure sine function has a period of oscillation of 2π, however, changing this value of ω can make the period of oscillation longer or shorter (more generally, ω=2π/T, where T is the period). If we assume that our time, t, is in hours, then we want a full period of oscillation of our temperature function to be 24 (since there are 24 hours in a day). From this we can find ω=2π/24 since we want a full 2π oscillation every 24 hours.
C is simply the average temperature, which is 60 degrees in this case. Hence, C=60.
Finally, we need to dermine the phase shift, Φ. We can solve for this phase shift with a simple equation since the question provides us that "the average daily temperature first occurs at 8 AM". Remember, we said the average temperature was 60 degrees, hence we are essentially saying the following mathematically:
T(8)=60=sin(8ω+Φ)+60 (Note: I won't plug in ω until the end to make the algebra easier)
we can solve for Φ with a few lines of algebra:
60=sin(ω(8)+Φ)+60
⇒sin(8ω+Φ)=0
⇒8ω+Φ=π (Since sin(x)=0 if x is a multiple of π and we are only interested when it first equals zero)
⇒Φ=π-8ω
⇒Φ=π-2π/3
⇒Φ=π(1-2/3)
⇒Φ=π/3
Hence your temperature function will take the form T(t)=sin((2π/24)t+π/3)+60
