One of the most important uses of trigonometry is in describing periodic processes. We find many such processes in nature: the swing of a pendulum, the tidal movement of the ocean, the variation in the length of the day
throughout the year, and many others.
All of these periodic motions can be described by one important family of functions, which all physicists use. These are the functions of the form.
where the constants a and k are positive, and f3 is arbitrary. In this chapter, we will describe their graphs, which we will call sinusoidal curves. Since they are so important, we will discuss them step-by-step, analyzing in turn
each of the parameters a, k, and b.
Graphing the basic sine curve
y = a sin k (x – b) for a = 1, k = 1, b = 0
In Chapter 5 we drew the graph of y = sin x:
Let us review how we obtained this graph. On the left below is a circle with unit radius. Point P is rotating around it in a counterclockwise direction, starting at the point labeled A.
The period of the function y = sin x
The period of the function y = sin x
for any number x.
Definition: A function f has a period p if f (x) = f (x + p) for all values of x for which f(x) and f(x + p) are defined.
Definition: The period of a periodic function .f (x) is the smallest positive real number p such that f(x + p) = f(x) for all values of x for which f (x) and f (x + p) are defined.
Let us also draw the graph of the function y = cos x. Following the same methods, we find that the graph is as shown below:
Periods of other sinusoidal curves y =a sin k(x – b) for a = 1, b = 0, k > 0
Having drawn one period, of course, it is easy to draw as much of the whole graph as we like (or have room for):
The graph is the same as that of y = sin x, but compressed by a factor of 3 in the x- direction. In general, we have the following result: Fork > 1, the graph of y = sin kx is obtained from the graph y = sinx by compressing it in the x-direction by a factor of k. What if 0 < k < 1? Let us draw the graph of y = sin x 15. Since the
our function takes on the same values as the function y = sin x, but stretched out over a longer period.
Again, we have a general result:
For 0 < k < 1, the graph of y = sin kx is obtained from the graph y = sinx by stretching it in the x-direction by a factor
The amplitude of a sinusoidal curve.
y=a sin k(x-,8); a>0, 8=0,k>0
Example 61 Draw the graph of the function y = 3 sin x.
Solution: The values of this function are three times the corresponding values of the function y = sin x. Hence the graph will have the same period, but each y-value will be multiplied by 3:
We see that the graph of y = 3 sin x is obtained from the graph of y = sin x by stretching in the y-direction. Similarly, it is not hard to see that the graph of y = ( 1 /2) sin x is obtained from the graph of y = sin x by a compression in the y-direction.
We have the following general result: For a > 1, the graph of y = a sin x is obtained from the graph
y = sinx by stretching in they-direction. For O < a < 1, the graph of y = a sin x is obtained from the graph y = sin x by
compressing in they-direction.
Analogous results hold for graphs of functions in which the period is not 1, and for equations of the form y = a cos x. The constant a is called the amplitude of the function y =a sin k(x- b).
Shifting the sine y =a sink(x- b); a= 1, k = 1, b arbitrary
We start with two examples, one in which f3 is positive and another in which b is negative.
graph of y = sin x. The positions of three particular points 1 on the original graph will help us understand how to do this:
Of course, with a calculator or a table of sines, you can get many more values. Or, if you have a good memory, you can remember the values of the sines of other particular angles. But these three points will serve us well for quite a while.
The graph of the function y = sin(x- b) is obtained from the graph of y = sin x by a shift of fJ units. The shift is towards the left if b is negative, and towards the right if b is positive.
The number fJ is called the phase angle or phase shift of the curve. Analogous results hold for the graph of y = cos(x – b).
Shifting and stretching
Graphing y =a sink(x- b)
We run into a small difficulty if we combine a shift of the curve with a change in period.
Some special shifts: Half-periods We will see, in this section, that we have not lost generality by restricting a and k to be positive, or by neglecting the cosine function.
It is useful to write our general equation as y = a sin k(x + y ), where y = – b. Then, for positive values of y, we are shifting to the left. For
In fact, we can state the following alternative definition of a period of a function:
A function y = f (x) has period p if the graph of the function coincides with itself after a shift to the left of p units.
Our original definition said that a function f(x) is periodic with period p if f(x) = f(x + p) for all values of x for which these expressions are defined.
Our new definition is equivalent to the earlier one, since the graph of y = f(x), when shifted to the left by p units, is just the graph y =f(x + p). These graphs are the the same if and only if f(x) = f(x + p).
Let us see what happens when we shift the graph y = sin x to the left
The graph of the function y = – sin x can be obtained from the graph y = sin x by a shift to the left of rr.
In fact, we do not need to make a separate study of the curves y=a sin k(x + y) for negative values of a. We need only adjust the value of y, and we can describe each such curve with an equation in which a > 0.
The following general definition is convenient:
The number p is called a half-period of the function f if f(x + p) = – f(x), for all values of x for which f(x) and f (x + p) are defined.
We have shown that n is a half-period of the function y = sin x.
Now let k = 3. We obtain the following graph:
Without loss of generality, we can take k to be positive.
The constant b is called the phase or phase shift of the curve. It tells us how much the curve has been shifted right or left. If we allow b to be arbitrary, we need not consider negative values of a or k, and we need not study separately curves expressed using the cosine function.
Graphing the tangent and cotangent functions
The function y = tan x is different from the functions y = sin x and y = cos x in two significant ways. First, the domain of definition of the sine and cosine functions is all real numbers. However, tan x is not defined for
Second, the sine and cosine functions are bounded: the values they take on are always between -1 and 1 (inclusive). But the function y = tanx takes on all real numbers as values.
These differences are easily seen in the graph of the function y = tan x:
An important question about sums of sinusoidal functions
We hope that from this material you have seen the importance, and the beauty, of the family of sinusoidal curves that we have been studying.
Physicists call this family the curves of harmonic oscillation.
Let us now consider the following question. Suppose we have two sinusoidal curves (harmonic oscillations):
Will the sum of these two also be a sinusoidal curve (harmonic oscillation)? That is, will
That is, the sum of two harmonic oscillations is again a harmonic oscillation if and only if the original frequencies are the same. The results of the next few sections will allow us to explore this situation.
Definition: If we have two functions f(x) and g(x), and two constants a and b, then the expression af (x) + bg(x) is called a linear combination of the functions f(x) and g(x).
Let us look at the graph of a linear combination of sinusoidal curves.
so there exists an angle a such that
Write each function in the form y = a sin k(x – b). What is the maximum value of each function?
3. y = sinx + cosx
4. y = sinx- cosx
5. y = 4sinx + 3cosx
6. y = sin2x + 3cos2x
7, 8: Write each function in the form A sinx + B cosx:
Linear combinations of sinusoidal curves with the same frequency
Now we are ready to address the important question of Section 9.
Theorem The sum of two sinusoidal curves with the same frequency is again a sinusoidal curve with this same frequency.
Proof Let us take the two sinusoidal curves
Note that the two functions we are adding may have different amplitudes.
The result depends only on their having the same period. This result is very important in working with electricity. Alternating electric current is described by a sinusoidal curve, and this theorem says that if we add two currents with the same periods, the resulting current will have this period as well. So if we are drawing electric power from different sources, we need not worry how to mix them (whether their phase shifts are aligned), as long as their periods are the same.
The next result is important in more advanced work:
Theorem If a linear combination of the functions y = sin kx and y = cos kx is shifted by an angle b, then the result can be expressed as a linear combination of the same two functions.
Proof Let us take the linear combination a sin kx + b cos kx and shift it by an angle b. The result is a sin k(x- b) +bcosk(x- b).
We know that cosk(x- b) can be written as sink(x -y), for some angle y.
Thus we can write our shifted linear combination as a sink(x- b) + b sink(x- y).
But this is a sum of sinusoidal curves with the same frequency k, so the previous theorem tells us that it can be written as a single sinusoidal curve with frequency k (even though the shifts are different!). And we know, from Section 9, that such a sum can be written as a linear combination of sin kx and cos kx.
Example 68 Suppose we take the graph of a linear combination of y = sinx and y= cosx:
y = 2 sin x + 4 cos x
which is again a linear combination of y = sin x and y = cos x.
This technique works whenever we apply a shift to a linear combination of y = sin kx and y = cos kx. The proof follows the reasoning of the above example.
A final comment: We have not considered linear combinations of sines and cosines with different frequencies: This is a more difficult situation, and leads to some very advanced mathematical topics, such as Fourier Series and almost periodic functions. We will return to this question a bit later.
Linear combinations of functions with different frequencies So far, we have some important results about linear combinations of sines and cosines with the same frequency. We would like to investigate the sum
This is particularly easy to see for those points where sin x = 0. For these points, the value of x + sin x is just x:
In between these points, the line y = x is lifted up slightly, or brought down slightly, by positive or negative values of sin x. We can think of the sine curve as “riding” on the line y = x.
Example 70 Graph the function y = sin x + 1 I 10 sin 20x.
Solution. This seems much more complicated, but in fact can be solved using the same method as the previous examples. We graph the two curves y = sinx andy = 1110 sin 20x independently, then add their y-values at
Again, we can think of one curve “riding” on the other. This time the curve y = 1 /10 sin 20x “rides” on the curve y = sin x, or perturbs it a bit at each point.
Note that our new curve is not a sinusoidal curve. We cannot express it either in the form y = a sin k(x- /3) or in the form y = A sin kx +b cos kx.
Finding the period of a sum of sinusoidal curves with different periods.
A discovery of Monsieur Fourier
Mathematicians say that this sequence of functions converges to a limit, and that this limit is the function whose graph is given above. In fact, this is a special case of the very important mathematical theory of Fourier series.
The French physicist Fourier discovered that almost any periodic function, including some with very complicated or bizarre graphs, can be represented as the limit of a sum of sines and cosines (the above example doesn’t happen to contain cosines).
He also showed how to calculate this sum (using techniques drawn from calculus).
Fourier’s discovery allows mathematicians to describe very simply any periodic function, and physicists can use these descriptions to model actions that repeat.
For example, sounds are caused by periodic vibrations of particles of air. Heartbeats are periodic motions of a muscle in the body.
These phenomena, and more, can be explored using the mathematical tools of Fourier analysis.
Many phenomena in nature exhibit periodic behavior: the motions repeat themselves after a certain amount of time has passed. The sine function, it turns out, is the key to describing such phenomena mathematically.
The following exercises concern certain periodic motions. Their mathematical representations remind us of the sine curve, but are not exactly the same.
In more advanced work trigonometric functions can indeed be used to describe these motions.
How to explain the shifting of the graph to your younger brother or sister
When we were little, we used to go every few months to the doctor. The doctor would measure our height, and make a graph showing how tall we were at every visit.
Here is the graph for my height
Two years later, when you were born, our parents asked the doctor if she could predict your growth year-by-year. Well, she couldn’t exactly do this, but she said: “If the new baby follows the same growth curve as your older child, then he will be as tall as the older one was three years earlier.”
So the doctor was predicting a growth curve for you which looks like this:
You will be 3 feet tall exactly two years after I was 3 feet tall, and 4 feet tall also exactly two years after I was, and so on. Your graph is the same as mine, but shifted to the right by two years. If you want to know the prediction for your height, just look at what my height was two years ago.
So if my graph is described by the equation height = f(year), then your graph is described by the equation height= f(year- 2).
Of course, it hasn’t quite turned out this way. My growth curve was not exactly the same as yours. So the doctor’s prediction was not accurate. But for some families, it is accurate.
Of course, when I was very little, I couldn’t stand up, so they measured my “length.” When I learned to stand, this became my height.
In just the same way, if you are dealing with the graph y = sin(x- a), rather than y = sin x, you must “wait” for x to get bigger by a before the height of the new graph is the same as that of the old graph. So the new graph is shifted a units to the right.
Sinusoidal curves with rational periods
We have taken, as our basic sine curve, the function y = sin x. The period of this function is 2rr, which is an irrational number. The other functions we’ve investigated also have irrational periods. Can a sine curve have a rational period?
The exercises below require the construction of sinusoidal curves with other rational periods.
From graphs to equations
A tale is told of the Russian tsar Alexei Mikhaelovitch, the second of the Romanov line (1629-76; reigned 1645-76). His court astronomer came to him one day in December, and told him, “Your majesty, from this day forth the number of hours of daylight will be increasing.”
The tsar was pleased. “You have done well, court astronomer. Please accept this gift for your services.” And, motioning to a courtier, he presented the astronomer with a valuable gemstone.
The astronomer enjoyed his gift and practiced his arts, until one day in June, when he again reported to the tsar. “Your highness, from this day forth the number of hours of daylight will be be decreasing.”
The tsar scowled. “What? More darkness in my realm?” And he ordered the hapless astronomer beaten.
Of course, the variation in the amount of daylight was not the fault of this astronomer, or any other astronomer. It is due to the circumstance that the earth’s axis is tilted with respect to the plane in which it orbits the sun. Because of this phenomenon, the days grow longer from December to June, then shorter from June to December.
What is interesting to us is the rate at which the number of hours of daylight changes. It turns out that if we graph the number of hours of daylight in each day, we get a sinusoidal curve:
Since this curve is high above the x-axis, we have shown it with the y-axis “broken,” so that you can see the interesting part of the graph. If you don’t like this, try redrawing the curve without a “broken” y-axis. You
will find that most of your diagram is empty. We will learn more about this curve in the following exercises.
Notice that the “average” number of daylight hours is the same for each graph. This “average” is given by the y-coordinates along the line around which the curve oscillates: On certain days of the year, at each location, the actual number of hours of daylight is the same as the average number. How does the time of year at which this average
is actually achieved vary from location to location?