Question

in many parts of the coutry the average temp of a particular area may be modeled by a sin usoidal function. According to NCDC the average temp of k city is 92.2 in july and 32.2 in january assume that the period is 12 what is the best model for when july is month 0 and when january is month 0

Answer 1

first take Jan as month 0 (also month 12)
then July is month 6 (note Dec is month 11)

n angle in radians with period 12
0Jan 0
1feb 1*2pi/12 = pi/6
2Mar 2*2 pi/12 = pi/3
3Apr 3*2pi/12 = pi/2
4may 4*2pi/12 = 2 pi/3
5Jun 5*2pi/12 = 5pi/6
6Jul 6*2 pi/12 = pi (half the period)
7Aug 7*2 pi/12 = 7 pi/6
8sept 8*2 pi/12 = 4 pi/3
9oct 9*2 pi/12 = 3 pi/2
10Nov 10*2pi/12 = 5pi/3
11Dec 11*2pi/12 = 11pi/6
12Jan 12*2pi/12 = 2 pi full circle
note - angle is n pi/6
so
min at n = 0
so
T = a - b cos (n pi/6)
32.2 = a - b cos 0 = a-b
92.2 = a - b cos pi = a+b

a-b = 32.2
a+b = 92.2
solve for a and b
then
T = a - b cos (n pi/6)

go through that again with n = 0 at July