1. Annual Temperature Cycles
2. Trigonometric Functions
3. Radian Measure
4. Sine and Cosine
5. Period, Amplitude, and Phase
6. Examples 
7. Worked Examples
8. Return to the Annual Temperature Variation
9. Degree and Radian Conversions
10. References

Many phenomena in biology appear in cycles. Often these cycles are driven by the natural physical cycles that result from the daily cycle of light or the annual cycle of the seasons. Oscillations are most easily studied using trigonometric functions. This section begins with a discussion of annual temperature variations, then we review some trigonometric functions.

Annual Temperature Cycles

Often the weather report states what the average expected temperature of a given day is. These averages are derived from long term collection of data on weather for a particular location. Clearly, there is a relatively wide variation from these averages, but they provide approximations to the expected weather for a particular time of year. The long term averages also provide a baseline to help researchers determine the predicts effects of global warming over the background noise of annual variation. Obviously, there are seasonal differences in the average daily temperature with higher averages in the summer and lower averages in the winter. Below we provide a table of showing the monthly average high and low temperatures for San Diego and Chicago.

What mathematical tools can help predict the annual temperature cycles? Polynomials and exponentials do not exhibit the periodic behavior that we see for these average monthly temperatures, so these functions are not appropriate for modeling this system. The most natural candidates for studying monthly temperatures are the trigonometric functions. Below are graphs of the average monthly temperatures for San Diego and Chicago, which are computed from the table by averaging the average high and low temperatures.

The two graphs above have some similarities and clear differences. They both show the same seasonal period as expected; however, the seasonal variation or amplitude of oscillation for Chicago is much greater than San Diego. Also, the overall average temperature for San Diego, being further south and near the ocean, is greater than the average for Chicago. The overlying models in the graph above use cosine functions. The fit using the cosine function provides a reasonable approximation though clearly there are errors due to other complicating factors in weather prediction. Before providing more details of the models for these temperature cycles, we review some basic facts about trigonometric functions.

Trigonometric Functions

The trigonometric functions are often called circular functions, which emphasizes their periodic nature and shows their connection to a circle. Let (x, y) be a point on a circle of radius r centered at the origin. Define the angle q between the ray connecting the point to the origin and the x-axis. (See the diagram below.) 

The six basic trigonometric functions are defined in terms of x, y, and r (including the sign of x and y) shown in the diagram above by the following:

We will concentrate almost exclusively on the first two of these trigonometric functions, sine and cosine.

Radian Measure

Before discussing the nature of the trigonometric functions in more detail, we need to discuss radian measure of the angle. If you have had trigonometry before, you probably used degrees to measure an angle. However, they are not the appropriate unit to use in Calculus. The easiest way to consider radian measure is to examine the unit circle (which is simply the circle in the diagram above with a radius of 1). From earlier courses, you may recall that the circumference of a circle is 2pr, so the distance around the perimeter of the unit circle is 2p. The radian measure of the angle q in the diagram above is simply the distance along the circumference of the unit circle. Thus, a 45^o angle is 1/8 of the distance around the unit circle or p/4 radians. Similarly, 90^o and 180^o angles convert to p/2 and p radians, respectively. Below are formulae for converting from degrees to radians or radians to degrees:

A javascript is provided for easy conversions of degrees to radians or radians to degrees through a hyperlink.

Sine and Cosine

From the formulae for sine (sin) and cosine (cos) above, we see that if you take a unit circle, then the cosine function gives the x value of the angle (measured in radians), while the sine function gives the y value of the angle. The tangent function (tan) gives the slope of the line (y/x). Below is a graph of the sine and cosine functions for the angles from -2p to 2p, i.e., the graph shows sin(x) and cos(x) for -2p < x < 2p. (Note that the x here is the angular measurement, which is the q argument above and not the x, which measures the adjacent side of the triangle.)

There are several things to notice about these graphs. First, you notice the 2p periodicity. In other words, the functions repeat the same pattern every 2p radians. This is clear from the circle above because every time you go 2p radians around the circle, you return to the same point. The second point is that both the sine and cosine functions are bounded between -1 and 1. The sine function has its maximum value at p/2 with sin(p/2) = 1 and its minimum value at 3p/2 with sin(3p/2) = -1. Because of the periodicity, these extrema reoccur every integer multiple of 2p.

Below is an applet to help you link the picture of the circle above and the trigonometric functions, sine and cosine, graphed above, using the dynamics of an applet. Click on the applet to place the point. The table below the applet gives the angle in degrees and radians and values of the sine and cosine functions at the angle chosen.

source code for trig6a.class

Below is a table of some important values of the trig functions to remember.

x	sin(x)	cos(x)
0	0	1
p/6	1/2
p/4
p/3		1/2
p/2	1	0
p	0	-1
3p/2	-1	0
2p	0	1

Period, Amplitude, and Phase

The sine and cosine functions above have a period of 2p and an amplitude of one, so how can we adjust these functions to fit other periodic data, such as the temperature data for Chicago and San Diego given in the introduction to this section? The discussion begins with a more general form of the sine function given below:

The value of A gives the amplitude of this function. The parameter w is often called the frequency of this function, and the period is given by 2p/w. The parameter f is called the phase shift. This shifts the curve f units to the right. (The quantity wf is called the phase angle .) By varying A and w, the sine function achieves different heights and periods, while f shifts the graph to the right or left. The cosine function can be similarly adjusted. In fact, there is a trigonometric identity that relates sine and cosine:

which says that the cosine function is exactly the same as the sine function but shifted out of phase by -p/2 (or the sine curve shifted to the left by p/2).

Example: Find the period and amplitude of

and sketch the graph.

Solution: From the information above, we see that the amplitude of this function is 4, which says that it oscillates between -4 and 4, and the period is p. The easiest way to find the period T is to let x = T in the function, then set the argument of the trig function to 2p, and finally solve this for T.

From our table above, this function begins at 0 when x = 0. It achieves a maximum of 4 at x = p/4, where the argument is p/2. The function decreases to a minimum of -4 at x = 3p/4, where the argument is 3p/2, then increases to where it completes its cycle at x = p. The graph is shown below for -2p < x < 2p.

Example: Sketch a graph of the function

for 0 < x < 2p. Give the period and amplitude of this graph.

Solution: This function is shifted vertically by the constant 3. The amplitude is the 2 multiplying the cosine function, but now the graph will oscillate between 1 < y < 5. The easiest way to determine the high and low points of the graph is to realize that the largest and smallest values of the cosine function are 1 and -1, respectively. By substituting these values in the expression for y, we obtain

To find the period, T, we solve

Thus, the period of this function is 2p/3. Below is a graph of this function.

As usual, there are more examples worked the Examples Section found through the hyperlink.

Return to the Annual Temperature Variation

At the beginning of this section, there is an example showing the temperature variation between the seasons for Chicago and San Diego. We want to show what the mathematical models are for the curves in the graphs and explain them in terms of the definitions listed above to give you a better intuitive feel for the period, amplitude, phase shift, and vertical shift. The models employed are of the form

where T is the average monthly temperature and m is the month with January = 0. We need to find the appropriate values for the parameters A, B, w, and f. The technique for fitting the actual data to the model employs a couple of techniques. First, we know that the period of this function must be 12 months. This constrains our parameter w to satisfy

References

[1] Rand McNally Map and Travel Information, 2000.