Students sometimes find this counterintuitive. It may help to think of it in these terms. If we graph , we have to move it back two units in order to transform it back to a “normal” cosine wave. For , we must move it to the right to return it to the correct place. The graph of is identical to that of , but for values that are two less than those of the original cosine function.
Example 2
Sketch
This is a sine wave that has been translated to the right.
Horizontal translations are also referred to as phase shifts. Two waves that are identical, but have been moved horizontally are said to be “out of phase” with each other. Remember that cosine and sine are really the same waves with this phase variation.
can be thought of as a cosine wave shifted horizontally to the right by , or .
Alternatively, we could also think of cosine as a sine wave that has been shifted to the left.
Functions with both horizontal and vertical translations
If we combine the two types of translations, the general functions become:
sine/cosine waves that have been translated units horizontally and units vertically.
Example 3
Draw a sketch of
This is a cosine wave that has been translated up and to the right. It may help you to use the quadrant angles to draw these sketches. If you plot the points of at , , , , (as well as the negatives), and then translate those points before attempting to draw the curve you will most likely get better results.
Example 4
Draw a sketch of
This is a sine wave that has been translated down. Think of the argument of the function as equivalent to so it is also being moved to the left.
Again, start with the quadrant angles on and translate them down .
Then, take that result and shift it , or , to the left.
Example 5
Write the equation of the following sinusoid:
Notice that you have been given some points to help identify the curve properly. Remember that sine and cosine are essentially the same wave so you can choose to model the sinusoid with either one. If we think of the function as starting on the axis at a maximum point, it is often easier to use the cosine function. The general formula is:
From the points on the curve, the first maximum point to the right of the axis occurs at halfway between and , or . In the next lesson we will combine these translations with changes in period and amplitude as well, but for now, because the next maximum occurs to the right of that, or at , there is no change in period in this function. This means we can think of this as a “normal” cosine wave that has been translated to the right, or . The vertical translation value can be found by locating the center of the wave. If it is not obvious from the graph, you can find the center by averaging the minimum and maximum values.
This center is the wrapping line of the translated function and is therefore the same as . In this example, the maximum value is and the minimum is . So,
Placing these two values into our equation gives:
Actually, because the cosine graph is periodic, there are an infinite number of possible answers for the horizontal translation. If we keep going in either direction to the next maximum and translate the wave back that far, we will obtain the same graph. Some other possible answers are:
Because sine and cosine are essentially the same function, we could also have modeled the curve with a sine function. Instead of looking for a maximum peak though, for sine we need to find the middle of an increasing part of the wave to consider as a starting point. Can you see why we usually use cosine? It is even difficult to describe!
The coordinates of this point may not always be obvious from the graph. It this case, the drawing shows that one of those points occurs at . So the horizontal, or value would be . The vertical shift, amplitude, and frequency are all the same as the were for the cosine wave because it is the same graph. So the equation would become:
And, once again, there are an infinite number of other possible answers if you extend away from the value multiples of in either direction. Here are two examples.
Lesson Summary
We can transform any sinusoidal function using a vertical or horizontal transformation. These transformations behave according to the following guidelines:
sine/cosine waves that have been translated units horizontally and units vertically.
Review Questions
For problems 1a-1e, find the equation that matches each condition.
Express the equation of the following graph as both a sine and a cosine function. Several points have been plotted at the quadrant angles to help.
For problems 3-7, match the graph with the correct equation.
Sketch the graph of on the axes below.
Review Answers
B
E
D
C
A
note: this list is not exhaustive, there are other possible answers.
C
D
A
B
General Sinusoidal Graphs
Learning Objectives
A student will be able to:
Given any sinusoid in the form: or , identify the transformations performed by and .
Graph any sinusoid given an equation in the form or .
Identify the equation of any sinusoid given a graph and some critical values.
Introduction
Now that we have covered the four basic transformations of sine and cosine graphs, students will combine them by finding equations and graphing waves that have undergone any combination of these various transformations.
The Generalized Equations
In the previous two sections, you learned how to translate and dilate sine and cosine waves both horizontally and vertically. Now you are ready to combine these transformations. If we put together all the constants that we have covered, the general equation of a sinusoid becomes:
where is the amplitude, is the frequency, is the vertical translation, and is the horizontal translation.
Remember also the relationship between period and frequency. The frequency is given in the equation as B and the period can be found given the formula:
If we are given the period and need to find the frequency, the formula becomes:
With this knowledge, we should be able to sketch any sine or cosine function as well as write an equation given a graph.
Drawing Sketches/Identifying Transformations from the Equation
Example 1
Given the function: :
a. Identify the period, amplitude, and frequency.
b. Explain any vertical or horizontal translations present in the equation.
c. Sketch the graph from to .
a. From the equation, the amplitude is and the frequency is also . To find the period we use:
So, there are two waves from to and each individual wave requires to complete.
b. and , so this graph has been translated up, and to the left.
c. To sketch the graph, start with the graph of
Then, translate the graph to the left (the value).
Next, move the graph up ( value)
No we are ready to tackle the dilations. Remember that we are considering the “starting point” of the wave to be because of the horizontal translation. A normal sine wave takes to complete a cycle, but this wave completes one cycle in . Where will this sine wave complete its cycle?
The first wave will complete at , then we will see a second wave from to and a third from to . There is also a complete wave from to . Start by placing points at these values:
Using symmetry, each interval needs to cross the line in the center.
One sine wave contains a “mountain” and a “valley”.
So the mountain “peak” and the valley low point must occur halfway between the points above.
Connect the points with a smooth curve.
Extend the curve through the domain.
; Finally, extend the minimum and maximum points to match the amplitude of .
Example 2
Given the function: :
a. Identify the period, amplitude, and frequency.
b. Explain any vertical or horizontal translations present in the equation.
c. Sketch the graph from to .
a. From the equation, the amplitude is and the frequency is . To find the period we use:
So, there is only one half of a cosine wave from to and each individual wave requires to complete.
b. and , so this graph has been translated up, and to the right.
c. To sketch the graph, start with the graph of
Adjust the amplitude so the cosine wave reaches up to and down to negative three. This affects the maximum points, but the points on the axis remain the same. These points are sometimes called nodes.
Many students think that one complete cosine wave has more of a shape.
According to the period, we should see one of these shapes every , or one-half over .
So this half of a wave needs to be spread symmetrically between and , which means it will cross the axis halfway through, or at . Plot these points.
Then connect them with a smooth curve.
Fill in the rest of the curve to .
Now, shift the graph to the right.
Finally, we need to adjust for the vertical shift by moving it up .
So, the completed graph will look like this:
Writing the Equation from a Sketch
In order to be able to write the equation from a graph, you need to be provided with enough information to find the four constants.
Example 3
Find the equation of the sinusoid graphed here.
First of all, remember that strictly speaking, both sine and cosine could be used to model these graphs. However, it is usually easier to use cosine because the horizontal shift is easier to locate in most cases.
Therefore, the model that we will be using is:
One of the first things that should jump out at you in this graph is that if we think of it as a cosine function, it has a horizontal translation of zero. The maximum point is also the intercept of the graph, so there is no need to shift the graph horizontally and therefore, is really .
The amplitude is the height from the center of the wave. If you can’t find the center of the wave just by sight, you can calculate it. The center should be halfway between the highest and the lowest points, which is really the average of the maximum and minimum. This value will actually be the vertical shift, or value.
In this case, the maximum is and the minimum is .
The amplitude is the height from the center line, or vertical shift, to either the minimum or the maximum. Since this distance is half of the total height, this can be calculated by taken the difference between the minimum and maximum values (the total height), and dividing it by .
The last value to find is the frequency. In order to do so, we must first find the period. The period is the distance required for one complete wave. To find this value, look at the horizontal distance between two consecutive maximum points.
On our graph, the period is , so
We have now calculated each of the four parameters necessary to write the equation.
Replacing them in the equation gives:
If we had chosen to model this curve with a sine function instead, the amplitude, period and frequency, as well as the vertical shift would all be the same. The only difference would be the horizontal shift. The sine wave starts in the middle of an upward sloped section of the curve as shown by the red circle.
This point intersects with the vertical translation line and is a third of the distance back to . So, in this case, the sine wave has been translated to the left. The equation using a sine function instead would have been:
Lesson Summary
The general equations for any sinusoidal function are:
where is the amplitude(vertical dilation), is the frequency(horizontal dilation), is the vertical translation, and is the horizontal translation.
The period and frequency exhibit an inverse relationship to each other such that:
and
Cosine and sine waves are really the same function, but are out of phase with each other. A cosine wave is usually considered to have a maximum value equal to the intercept, but once you allow for horizontal translations any sinusoid could be considered to be either sine or cosine. When finding the equation of a sinusoidal graph, it is often easier to use cosine in the equation if you are given the coordinates of the maximum and/or minimum points. The horizontal shift for a cosine model is the coordinate of the first maximum peak to the right of the axis. The period is the horizontal distance between two consecutive maximum points.
The vertical shift, or value, can be found by averaging the maximum and minimum points. The amplitude, or value, can be found by subtracting the minimum from the maximum and dividing by .
Review Questions
For problems 1 through 5, identify the amplitude, period, frequency, maximum and minimum points, vertical shift, and horizontal shift.
Review Answers
This is a sine wave that has been translated to the right and up. The amplitude is and the frequency is . The period of the graph is . The function reaches a maximum point of and a minimum of .
This is a sine wave that has been translated down and to the left. The amplitude is and the period is . The frequency of the graph is . The function reaches a maximum point of and a minimum of .
This is a cosine wave that has been translated up and to the right. The amplitude is and the frequency is . The period of the graph is . The function reaches a maximum point of and a minimum of .
This is a cosine wave that has not been translated vertically. It has been translated to the left. The amplitude is and the frequency is . The period of the graph is . The function reaches a maximum point of and a minimum of . The negative in front of the cosine function does not change the amplitude, it simply reflects the graph across the axis.
This is a cosine wave that has been translate up and has an amplitude of . The frequency is and the period is . There is no horizontal translation. Putting a negative in front of the value reflects the function across the axis. A cosine wave that has not been translated horizontally is symmetric to the axis so this reflection will have no visible effect on the graph. The function reaches a maximum of and a minimum of .
other answers are possible given different horizontal translations of sine/cosine
or
Chapter 3: Trigonometric Identities
Fundamental Identities
Reciprocal, Quotient, Pythagorean
The three fundamental trigonometric functions are sine, cosine and tangent, and can be defined in terms of an angle, , in a right triangle.
At times during this chapter and beyond, it may be useful to know the reciprocals of these basic functions.
The three fundamental reciprocal trigonometric functions are cosecant , secant and cotangent and are defined as:
Using the fundamental trig functions, sine and cosine and some basic algebra can reveal some interesting trigonometric relationships. Note when a trig function such as is multiplied by itself, the mathematical convention is to write it as . [ can be interpreted as the sine of the square of the angle, and is therefore avoided.]
OR:
Using the notation from the diagram above, this calculation is:
and , so and
or
By the Pythagorean Theorem
So Therefore
This is known as the Trigonometric Pythagorean Theorem.
Alternative forms of the Theorem are:
The second form is found by taking the first form and dividing each of the terms by , while the third form is found by dividing all the terms of the first by .
If the sine of the angle is divided by the cosine of the angle or (opposite leg/hypotenuse)/(adjacent leg/hypotenuse), the result will equal (opposite le
g)/(adjacent leg) and that is also equal to the tangent of the angle. Or, using the notation from the picture above,
since and , then or
, and since .
This final statement, , is an important trigonometric identity, as well as the its reciprocal,
There is another way to look at the tangent function besides (opposite leg)/(adjacent leg). For example by knowing the tangent function is equivalent to provides insight that the tangent function cannot be defined at , since the cosine of is zero. By knowing alternative forms of a trigonometric function or a trigonometric expression, students can have a better understand of the behavior of these functions.
In summary, these two forms of tangent and cotangent are:
Note since and , then or
, and since
CK-12 Trigonometry Page 11
