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Chapter 8 Trig Functions and Sinusoidal Function s

We start by reviewing trig functions of an angle measured in radians. Because this is review material, we will go through it rapidly.

We first draw the unit circle in the xy-plane. The standard way of measuring an angle is to start with the positive x-axis and rotate counterclockwise. As we go through the angle, we sweep around the circumference of the unit circle. The radian measure of the angle is defined to be the distance around the circumference that we travel. That is, a full revolution of is equal to radians, half a revolution is = radians, and so on. Other frequently used equivalences are: rad, rad, rad, rad. All angles in this course will be assumed to be measured in radians unless otherwise stated .
The designation ``radian'' will often be omitted, e.g., if we write ``an angle of ,''` we mean a angle.

By a negative angle we mean a clockwise angle . For example, an angle of (i.e., ) brings us to the same position as an angle of (which is ).

In general, if we have a circle of radius and sweep out an angle of radians, then the distance around the circumference through which we have traveled is simply . Of course, one full circumference is , corresponding to rotation through radians.

Returning to the unit circle (see the drawing below), after rotating through an angle of radians let ( ) be the point where we end up on the circumference. Then the six trig functions: sine (sin) , cosine (cos), secant (sec), cosecant (csc), and tangent (tan) and cotangent (cot) are defined as follows:

, ,

With these we have the familiar relations (identities):

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From the definition of we see that as increases from to i (i.e., as we go once around the unit circle , goes from to to to and back to . Then as goes from to (i.e., as we go once again around the circle), the same values of are repeated. We say that the function has period , i.e., we have .

Here are some other relations for the trig functions that follow easily from the above definitions: , , , , , , , [Maple Math] , and . The last of these relations is familiar from high school trigonometry: the cosine of an angle is equal to the sine of the complementary angle .

The identity [Maple Math] can also be seen from our definitions as follows: the x-coordinate of the point on the unit circle you arrive at when you rotate through counterclockwise starting at the positive x-axis is the same as the y-coordinate of the point you arrive at when you rotate through clockwise starting at the positive y-axis.

Sometimes one also needs the following formulas for the trig functions of a sum of two angles: , and . As a special case we have

the double angle formulas :

and .

This will be further explained in class

Also notice the sign of the trig functions in the different quadrants: is positive in the and quadrants (for angles between 0 and ), is positive in the and , and is positive in the and .

Graphs of Trig Functions

The most important graphs to be familiar with are the graphs of the functions , , and . (We're now using for the angle in radians; no longer denotes a coordinate of a point on the unit circle.) For certain the trig functions have easily stated exact values, namely, when is a multiple of or of . Here it is useful to draw the triangle with hypotenuse and the triangle with hypotenuse . Using these triangles, we arrive at a table of values of the trig functions, from which we can sketch the graphs that follow.

code for table and triangles

code for graphs

Sinusoidal Functions

A sinusoidal function is a function made up from the function (or , if time is playing the role of the x-variable) by inserting constants in various places. The reason for inserting these constants is that the pure sine function almost never occurs in the real world, but curves with a sine-like appearance (this is what the word ``sinusoidal'' means) arise frequently.

At this point you should review the material at the end of the section on functions: horizontal and vertical shifts, horizontal and vertical expansion, and the last batch of homework problems in that section.

Example 8.1: Write the cosine function as a shift of the sine function.
The graphs show that the cosine function is obtained by shifting the sine function to the left. Thus, = . This relation can also be obtained using the identities listed above: = .

We now give the general form for a sinusoidal function y = f(x). We start with y = sin x.

Step 1. Expansion: Vertically, we expand by a factor of , which is called the amplitude . We do this by replacing by . This makes the function go to a maximum of and a minimum of (rather than a maximum of and a minimum of as is the case for ). Next, we expand horizontally in such a way that the function repeats its pattern from to (rather than from to as with ). is called the period . That is, we want to expand horizontally by a factor of We do this by replacing x by inside the sine. To summarize, the result of the vertical and horizontal expansion is to replace by the function . The graph of this new function is shown below. Notice that as x goes from to , the value of the function goes from to to to and back to .

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Step 2. Shifting: We now want to move the whole sine wave upward a certain amount vertical shift and over to the right a certain amount (called the phase shift ). We do this by adding outside the function, and replacing by inside the function. The result is:

[Maple Math] .

In the graph of this function below, we indicate the meaning of the four constants A, B, C, and D.

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Example 8.2: Suppose that the mean daily temperature (i.e., the temperature in averaged over the 24-hour period) on the day of the year in Seattle is tabulated over a period of many years, and the results for each day are averaged. This is what we call the ``normal mean temperature " on the day. For simplicity, suppose that the year consists of months of days each. Take to be January 1, and so on, until for December . Further assume: (1) the normal mean temperature function is sinusoidal; (2) the coldest mean temperature is on January ; and (3) the warmest mean temperature is on July . Find a formula for . We graph by first plotting the two points , and (i.e., July ), y = . We then draw a sine wave in such a way that its minimum is at the first point, its maximum is at the second, and its period is a year, i.e., days. We also draw a horizontal dotted line half-way between the coldest and warmest temperature, i.e., at y = .

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Now we read the four constants off the graph. The amplitude A is 14, the period B is 360, and the vertical shift (the distance up to the dotted line) is 51. To find C we want to know when the curve crosses the dotted line on its way from a minimum to a maximum, i.e., when the curve looks like sin x looks when it crosses the origin. This can be found by taking the t-value half-way between the coldest day (which is t = 30) and the warmest day (which is t = 210). This t-value is (30+210)/2 = 120 (i.e., April 30). Thus, the phase shift is C = 120, and our equation is

y(t) = [Maple Math]

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