Trigonometric Functions
Explanation
Many equations involve Trigonometric functions when dealing with oscillating curves. For example the temperature of a specific location changes as the year goes round but continues to go from low in the winter to high in the summer. There is no simple way to find a derivative other than just remembering the short cut. On this page we will be using only a sine function and a cosine function so those are the only derivative short cuts that we will need however we will list others for reference.
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Trigonometric Derivatives
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Examples
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Real World Examples
The exact time that the sun sets each day changes as the seasons change and how the earth revolves around the sun. For example the days are longer in the summer while the nights are longer in the winter. Below are two graphs with plotted points where x is the beginning of the month and y is the time of sunset. There is also a equation that is supposed to match the data as closely as possible. In addition each equation is a transformation of either a sine function or a cosine function. The point at finding the derivative for these equations is to know if the time of sunset is getting earlier or later which all depends on the month.
Sunset Times For Boston
Above to the left is the equation to g(x) and to the right is how one finds the derivative of g(x). To find the derivative one needs only know the derivative of a sine function and the chain rule. Now suppose one wants to know the mathematical significance about the months 7th month and 12th months because that is where the graphs switches direction so they plug it into the derivative equation.
Both of these derivative values are close to zero. This implies that when the line switches direction the derivative becomes zero before the switch. From this one can come to understand that the zeros represent the lowest and highest points in this graph. The highest and lowest points represent the longest and shortest days of the year where the amount of sunlight hours per day slowly stops decreasing or slowly stops increasing to go back the other direction. Coincidentally these are the solstices which happen to be on June (9 days away from the 7th month) and December 21st (the 12th month).
Sunset Times For San Francisco
Above to the left is the equation to g(x) and to the right is how one finds the derivative of g(x). To find the derivative one needs only know the derivative of a sine function and the chain rule. Above we found the solstices but what about the equinoxes? They would be the points that are right in the middle, half way between the high points and the low points. If we assume that the solstices are in the 6th and 12th months then the equinoxes are in the 3rd and 9th months. So what would the derivatives be at those points?