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

Question

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?

Answer 1

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

Answer 2

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)

Answer 3

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

Answer 4

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)

Answer 5

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