Trying to decypher something, and am getting fouled up with the radians definition of periodicity.

Problem: Average temperatures vary over a one year period, or 365 days. Highest temperature = 5 C, lowest = -37 C. Lowest temperature is on day = 20, highest temperature is on day = 200

Write the cosine approximation of the equation for each day as T(d)= A+B(cos(C(d) + D).

OK. My solution, so far.
So the range = 5 --37 = 42.
Amplitude = B = 42/2 = 21
D = -20. (The low point is at day 20)
A = (5+-37)/2 = -32/2 = -16
But I can't logic out C properly. Period HAS to = 365, which is 2*pi radians, but then how do you get C from that?

when t = T, the argument must be 2 pi so

cos ( 2 pi t/T + phase )
= cos (2 pi t/365 + phase)

the low point of the cosine function is when the angle inside is pi radians because cos(pi) = -1
so
2 pi (20)/365 + phase = pi
.11 pi + phase = pi
phase = .89 pi

so we have
cos ( 2 pi t/365 + .89 pi)
or
cos (.017 t + 2.8)

so when t = 20, then 2 pi t/365

when t = T, the argument must be 2 pi so

cos ( 2 pi t/T + phase )
= cos (2 pi t/365 + phase)

the low point of the cosine function is when the angle inside is pi radians because cos(pi) = -1
so
2 pi (20)/365 + phase = pi
.11 pi + phase = pi
phase = .89 pi

so we have
cos ( 2 pi t/365 + .89 pi)
or
cos (.017 t + 2.8)

That .89 pi or 2.8 radians is your D by the way

So I was throwing my "logic" off by assuming that I needed to subtract off the 20 days offset as "D", right?

Instead of 2.8 (or 2.796)

And thank you very much, for your time and effort answering the above question.

To determine the value of C in the equation for each day as T(d) = A + B(cos(C(d) + D)), we need to understand the relationship between the day and the cycle of the temperature variation.

In this case, the period of the cycle is one year or 365 days. This corresponds to a full revolution in radians, which is equal to 2π.

To find C, we need to relate the period (365) to 2π. We can use the formula for converting between periods and radians:

C = 2π / period.

In our case, C would be:

C = 2π / 365.

This gives us the value of C that represents the rate at which the temperature cycle completes one revolution in radians over the 365-day period.

To summarize:

- A: -16 (average or mid-range temperature)
- B: 21 (amplitude or half the temperature range)
- C: 2π / 365 (rate of temperature cycle completion in radians per day)
- D: -20 (displacement or phase shift, since the lowest temperature occurs on day 20)

With these values, you can now approximate the temperature for any given day using the equation T(d) = A + B(cos(C(d) + D)).