## Modelling Seasonal Temperatures With the Sine Function

Sine functions are useful for modelling some periodic phenomena. The temperature varies with the seasons, and the variation has a period of approximately one year.

The average maximum temperature (°C) for Cape Town in each month is shown in the table below.

Month | January | February | March | April | May | June | July | August | September | October | November | December |

Mean Temperature°C | 28 | 27 | 25.5 | 22 | 18.5 | 16 | 15 | 16 | 18 | 21.5 | 24 | 26 |

The data is plotted below.

The data may be modelled by the equation

The period is 12 months so

is the average annual temperature, on which the seasonal temperature causes the variation, and may be found as the average of the maximim and minimum of the maximum monthly temperatures.

The amplitude A is half the difference between the maximum and minimum teperatures:

We have arrived at

The pointon the graph above is displaced 10 months from the origin – that is ifthen would be at the origin, so that C=10. We have

The model is superimposed on the graph in the diagram below. It is a very good fit.