CK-12 Trigonometry

by CK-12 Foundation

  Example 1

  Look at the following periodic graph and assume that the graph continues in the same way for all values of .

  No matter where you start on this graph, it will eventually repeat the same behavior. For example, if we start at the s-axis one period of the graph looks like this:

  This shape is along the axis, so we would say that this is a periodic function and the period of the graph is . The period is defined as the distance required to complete one cycle.

  Example 2

  Identify the period of the following graph and draw one period.

  This graph repeats itself every . Defining the shape of one period depends on where you start in the cycle. Imagine if you turn on your mp3 player, start a playlist, and place it on continuous play If you put on the earphones and listen to it for a while, it doesn’t matter when you happen to start listening, if you listen long enough, you will hear all of the songs in the list. If you set it down and pick it up to listen again later, you will still hear the same songs in the same sequence, just at a different starting point. So, here are a few possible ways to view one period of this graph.

  In this first example, we could start at . The period would then finish at . The next example could start at and finish at .

  Or, in this final example, we could start at and finish at .

  Graphing y = sin(x) and y = cos(x) on a graphing calculator

  To graph the basic sine and cosine waves on a graphing calculator, enter the functions in the y = menu.

  In this case the graphing style has been changed on the cosine graph to help identify the different graphs.

  Make sure you check the mode to be sure it is in radian measure.

  Next, we need to set a window that would be appropriate. Because the period is , it is very common to graph two entire periods by showing from to . When entering the window values, you can actually type and the calculator will estimate it for you.

  In order to mark the quadrant angles, make the xscl . The axis settings of to make the aspect ratio of the screen very close to being correct.

  Note again that sine and cosine are essentially the same graph. If you were to push either graph over , it would be identical to the other. We call this being “out of phase” with each other and we will investigate this further in the next lesson. For now, just remember that the cosine function has a intercept of and the sine a intercept of . Because of this, we use the general term “sinusoid” to describe the graphs of both sine and cosine.

  Amplitude

  The amplitude of a wave is basically a measure of its height. Because that height is constantly changing, amplitude is defined as the farthest distance the wave gets from the center of the wave. In a graph of , the wave is centered on the axis and the farthest it ever strays (in either direction) from the axis is .

  So the amplitude of (and for that matter) is .

  Period and Frequency

  We defined the period earlier as the horizontal distance needed before the values begin their periodic repetition. For both the graphs of and , the period is . As we learned in section 3, after completing one rotation of the unit circle, these values are the same.

  Frequency is a different measure that is related to period. In science, the frequency of a sound or light wave is the number of complete waves for a given time period (like seconds). In trigonometry, because all of these periodic functions are based on the unit circle, we usually measure frequency as the number of complete waves every . Because the sine and cosine graphs have exactly one complete wave over this interval, their frequency is .

  Period and frequency are inversely related. That is, the higher the frequency (more waves over ), the lower the period (shorter distance on the axis for each complete cycle).

  Period and Frequency of other Trigonometric Functions

  Here is the graph of tangent from section 3.

  Notice that one period of this graph looks like this:

  This occurs over a horizontal distance of , so the period of is . If we were to graph this function over , we would see complete tangent waves, so the frequency is .

  y = csc(x)

  Look at the graph of the cosecant.

  What would you consider to be one period of this wave? It really depends on where you begin, but it does take for the graph to repeat itself. A full period is one “valley” and one “mountain” shape together.

  So, just like the sinusoids, the cosecant function has a period of and a frequency of .

  Identifying the period and frequency of the secant and cotangent functions will be left to you in the exercise section.

  Transformations of Sine and Cosine Graphs: Dilations

  Change of Amplitude,

  Once you understand the basic features of the graphs of sinusoids, we can begin to learn how to alter their graphs. Recall how to transform a simple linear function like . By placing a constant in front of the value, you may remember that the slope of the graph affects the steepness of the line:

  The same was true of the basic parabolic function, . By placing a constant in front of the (you may have used the variable ), the graph would be either “steeper” or “flatter” in terms of the rate at which it grew (or decreases). In transformational geometry terms, we call this a dilation. A dilation is a stretching or shrinking of the graph that distorts the graph proportionally. So a function such as , has the same parabolic shape but it has been “squeezed” flatter so that it increases or decreases at a lower rate than the graph of . No matter what the basic function; linear, parabolic, or trigonometric, the same principle holds. If you want to dilate the function, multiply the function by a constant. Constants greater than will stretch the graph out vertically and those less than will shrink it vertically.

  Look at the graphs of and .

  Notice that the amplitude of is now . An investigation of some of the points will show that each y value is twice as large as those for . A look at a table of values with our special angles will show this numerically.

  angle

  Multiplying values less than will decrease the amplitude of the wave as in this case of the graph of :

  So, in general, the constant that creates the dilation is the amplitude of the sinusoid.

  Change of Period/Frequency, y = sin(Bx), y = cos(Bx)

  After observing the transformations that result from multiplying a number in front of the sinusoid, it seems natural to look at what happens if we multiply a constant inside the argument of the function, or in other words, by the value.

  For example, look at the graphs of and

  As you can see, we have increased the number of waves in the same interval. There are now waves over the interval from to . Consider that you are doubling each of the values, so when you plug in , for example, the argument of the function becomes . So the portion of the graph that normally corresponds to on the axis, now corresponds to half that distance—so the graph has been “scrunched” horizontally. Here is the table of values:

  angle

  Notice that the values at the end of each complete wave have been highlighted, so you can see that the graph of completes a wave every , or two complete waves from to or . The frequency of this graph is therefore , or the same as the constant we multiplied in the argument. The period (the distance for each complete wave) is .

  Example 3: What is the frequency and period of ?

  If we follow the pattern from the previous example, multiplying the angle by should result in the sine wave completing a cycle three times as often as . So, there will be three complete waves if we graph it from to . The frequency is , and if there are complete waves in , one wave will take a third of that distance, or . Here is the graph:

  This number that is multiplied by the angle (it is called ), will create a horizontal dilation. The larger the value of , the more compressed the waves will be horizontally. To stretch out the graph horizontally, we would need to decrease the frequency, or multiply by a number that is less than . Remember that this dilat
ion factor is inversely related to the period of the graph.

  In general:

  Where is the frequency, and the period is equal to .

  Example 4: What is the frequency and period of ?

  Using our generalization above, the frequency must be and therefore the period is , which simplifies to:

  Thinking of it as a transformation, the graph is stretched horizontally. We would only see of a complete wave if we graphed the function from to . To see a complete wave, therefore, we would have to go four times as far, or all the way from to .

  Changes of period, amplitude, and frequency

  If we generalize and allow for both horizontal and vertical dilations at the same time, the equations would become:

  where is the amplitude, is the frequency, and the period is .

  Example 5: Find the period, amplitude and frequency of and sketch a graph from to .

  This will be a cosine graph that has been stretch both vertically and horizontally. It will now reach up to and down to , and we would need to graph it all the way out to in order to see a complete period of the cosine wave. Since we are only going out to , we will only see half of a wave. A complete cosine wave looks like this:

  so half of it is this:

  We need to stretch this out so it finishes at , which means that at , or halfway, the graph should cross the axis:

  The final sketch would look like this:

  Example 6: Identify the period, amplitude, frequency, and equation of the following sinusoid:

  The amplitude is . Notice that the units are not labeled in terms of in this example. This appears to be a sine wave because the intercept is . Remember however, that sine and cosine are essentially the same, so in the next section when we learn to translate the graph horizontally, we will be able to treat it as a cosine wave as well.

  One wave appears to complete in (not !), so the period is . If one wave is completed in , how many waves will we see in ? In previous examples, you were given the frequency and asked to find the period using the following relationship:

  Where is the frequency and is the period. With just a little bit of algebra, we can transform this formula and solve it for :

  Because these are inverse relationships, we can simply interchange the values.

  So, the frequency is:

  If we were to graph this out to we would see (or a little more than ) complete waves.

  Replacing these values in the equation gives:

  Lesson Summary

  A periodic function is one in which the function repeats the same values over a given interval, or period. Because they are based on the repetition of rotations around the unit circle, all trigonometric functions are periodic. The frequency of a periodic function is the number of waves over a given interval, or for trigonometric functions. Sine and cosine are similar periodic functions that are called sinusoids. The amplitude of a sinusoid is the height of the wave, measured from its center. For and , the amplitude is , the frequency is , and the period is . We can transform the sinusoids using a vertical or horizontal dilation. These transformations behave according to the following guidelines:

  where is the amplitude and is the frequency.

  The period and frequency are inversely related by the following equations:

  , or where is the period and is the frequency.

  Review Questions

  Using the graphs from section 3, identify the period and frequency of and .

  Identify the minimum and maximum values of these functions.

  How many real solutions are there for the equation over the interval ?

  For each equation, identify the period, amplitude, and frequency.

  Given each of the sinusoids that follow:

  -identify the period, amplitude, and frequency.

  -write the equation.

  For each equation, draw a sketch from to .

  Review Answers

  min: , max:

  min: , max:

  min: , max:

  there is no minimum or maximum, tangent has a range of all real numbers

  min: , max:

  min: , max:

  d.

  period: , amplitude: , frequency:

  period: , amplitude: , frequency:

  period: , amplitude: , frequency:

  period: , amplitude: 2, frequency: 3

  period: , amplitude: , frequency:

  period: , amplitude: , frequency:

  period: , amplitude: , frequency: ,

  period: , amplitude: , frequency:

  period: , amplitude: , frequency:

  period: , amplitude: , frequency: ,

  Translating Sine and Cosine Functions

  Learning Objectives

  A student will be able to:

  Translate sine and cosine functions vertically and horizontally.

  Identify the vertical and horizontal translations of sine and cosine from a graph and an equation.

  Introduction

  In this lesson students will apply the general concepts of translation for any function to the sine and cosine functions. Both horizontal and vertical translations will be reviewed and then generalized to apply to any sinusoid.

  Vertical Translations

  When you first learned how to do vertical translations in a coordinate grid, you most likely started with simple shapes. Here is a rectangle:

  If we would like to translate this rectangle vertically, we simply would move all points and lines up by a specified number of units. We do this by adjusting the coordinate of the points. So to translate this rectangle up, we would simply add to every coordinate.

  This process worked the same way for functions. Since the value of a function corresponds to the value on its graph, to move a function up five units, we would increase the value of the function by . Here is the graph of the parabola, .

  To translate this function up five units, we increase the value by . Because is equal to , then the equation , should show this translation.

  In general, anything that we graph will be translated when we increase the value of the function by a constant. If we have any random shape, let’s call it a “blob,” adding a constant to the equation will move it up, and subtracting a constant will move it down.

  So, the graphs of and follow the same rules. That is, the graph of will be the same as , only it will be translated, or shifted, up.

  To help avoid some confusion, we will write this translation in front of the function: .

  To translate a cosine wave down then, we would write the function as:

  This would be a cosine wave with an amplitude of and a frequency of that has been shifted down.

  Various texts use different notation, but we will use as the constant for vertical translations. This would lead to the following equations:

  where is the vertical translation.

  There is another way to view this. Think of a sine or cosine wave as like a “snake” wrapped around a pole. For and , the graphs are wrapped around the axis, or the horizontal line, .

  For , we have already learned that it should be a “normal” sine wave that has been translated up . In this context though, let’s think of it as a sine wave that is wrapped around the line, .

  To generalize,

  are sine/cosine waves wrapped around the line, .

  Example 1

  Find the minimum and maximum of

  This is a cosine wave that has been shifted down , or is now wrapped around the line . Because it still has an amplitude of , the cosine wave will extend one unit above the wrapping line and one unit below it. The minimum is and the maximum is .

  Horizontal Translations (phase shift)

  Horizontal translations are a little less intuitive. If we return to the example of the parabola, , what change would you make to the equation to have it move to the right or left? Many students instinctive guess that if we move the graph vertically by adding to the value, then we should add to the value in order to translate horizontally. This is essentially corre
ct, but behaves in the opposite way than what you may think.

  Here is the graph of .

  Notice that adding to the value appears to have shifted the graph to the left.

  Sure enough, the graph moves the graph to the right.

  Let’s use the letter to represent the horizontal shift value. If this is the case, then subtracting from the value will shift the graph to the right. Adding can be thought of as subtracting the opposite of . So, will move the graph d units to the left.

  So, the sine and cosine functions follow this general rule:

  are sine/cosine waves that have been translated units horizontally.