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Answer on Trigonometry Question for cap

Question #5499
in math if the maximum # of people at a resort is 15000 and it drops to the minimum if 500 within 6 months what is the cosine equation that describes this function?
Expert's answer
Let's find the equation in a following form:

N(t) = A*sin(a(t+c)),

where N is the number of people and t is time in months.

Let's solve the following system of equations:

A*sin(a(0+c)) = 15000,
A*sin(a(6+c)) = 500;

Since |sin(x)| <= 1, we'll assume A = 15000.

sin(a(0+c)) = 1,
sin(a(6+c)) = 1/30;

sin(ac) = 1,
sin(6a+ac) = 1/30;

ac = pi/2,
6a+ac = arcsin(1/30);

6a+pi/2 = arcsin(1/30);

a = (arcsin(1/30)-pi/2)/6 ≈ -0.244342861570884;

c = pi/(2a) = pi/((arcsin(1/30)-pi/2)/3) = (3pi)/(arcsin(1/30)-pi/2) ≈ -6.428656506256111.

So, we've got the following model:

N(t) = 15000*sin(-0.244342861570884*(t-6.428656506256111)).

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