Sine and Cosine Functions

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature during the day are 63 and 37 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

Time past midnight in hours 0 6 12 18 24

Temperature 50 37 50 63 50

Notice that temperature fluctuates ± 13 degrees around 50.

24 hours is a full daily cycle. One cycle of a sinusoidal function is 2π radians, so we need the argument of the sin to be 2π when t = 24. Let the argument be (π/12) t. Using this:

D = 50 - 13sin{(π/12)t}.

Note the average temperature is 50. The average of any sinusoidal function over one cycle is zero. Thus 50 must be added to the sinusoid.

You can confirm this is the correct formula by inserting values of time of 0, 6, 12, 18, 24. The give temperatures of 50, 37, 50, 63, 50

Answer 2

The midline is the centerline about which the sine or cosine wave oscillates. To find the midline, add the max and min and divided by 2. So (63+37)/2 = 50. This happens to be the temp at midnight meaning at t = 0, the curve passes through this "starting point". A sine wave starts at the midline so this would be the easiest function to use. However, a positive sine wave goes up from the starting point. In this case, the temperature would drop after midnight to the minimum point in the early morning so we would need to use a negative sine wave. So our baseline would be:

D(t) = -Asin[B(x - C)] + D

where:

A is the amplitude (distance from the midline to either the max or min). In this case, A = 13 because the max temp is 13° higher than the midline and the min temp is 13° below the midline.

B is 2π/period. In this case the period is 24 hours since the max temp (or min) occurs once a day (every 24 hours). Since the period is 24, the B-value is 2π/24 or π/12.

C is the phase shift. Since the sine wave normally starts on the midline at t = 0 then C = 0 in this case.

D is the midline so D = 50 in this case.

Put these together to get:

D(t) = -13sin[(π/12)x] + 50