The negative value of results in a reflection across the x -axis of the sine function , as shown in Figure. Now that we understand how and relate to the general form equation for the sine and cosine functions, we will explore the variables and Recall the general form:. The value for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function. If the graph shifts to the right. If the graph shifts to the left.
The greater the value of the more the graph is shifted. Figure shows that the graph of shifts to the right by units, which is more than we see in the graph of which shifts to the right by units. While relates to the horizontal shift, indicates the vertical shift from the midline in the general formula for a sinusoidal function. The function has its midline at. Any value of other than zero shifts the graph up or down.
Figure compares with which is shifted 2 units up on a graph. Determine the direction and magnitude of the phase shift for. In the given equation, notice that and So the phase shift is. We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before Therefore can be rewritten as If the value of is negative, the shift is to the left.
Determine the direction and magnitude of the vertical shift for. In the given equation, so the shift is 3 units downward. Given a sinusoidal function in the form identify the midline, amplitude, period, and phase shift. Determine the midline, amplitude, period, and phase shift of the function. Next, so the period is.
There is no added constant inside the parentheses, so and the phase shift is. Inspecting the graph, we can determine that the period is the midline is and the amplitude is 3. Determine the formula for the cosine function in Figure. To determine the equation, we need to identify each value in the general form of a sinusoidal function. When the graph has an extreme point, Since the cosine function has an extreme point for let us write our equation in terms of a cosine function.
We can see that the graph rises and falls an equal distance above and below This value, which is the midline, is in the equation, so. The greatest distance above and below the midline is the amplitude. The maxima are 0. So Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so Also, the graph is reflected about the x -axis so that.
The graph is not horizontally stretched or compressed, so and the graph is not shifted horizontally, so. Determine the formula for the sine function in Figure. Determine the equation for the sinusoidal function in Figure. With the highest value at 1 and the lowest value at the midline will be halfway between at So.
The distance from the midline to the highest or lowest value gives an amplitude of. The period of the graph is 6, which can be measured from the peak at to the next peak at or from the distance between the lowest points. Therefore, Using the positive value for we find that. So far, our equation is either or For the shape and shift, we have more than one option. We could write this as any one of the following:.
While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes. Again, these functions are equivalent, so both yield the same graph. Write a formula for the function graphed in Figure. Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs.
Now we can use the same information to create graphs from equations. Given the function sketch its graph. Sketch a graph of. The quarter points include the minimum at and the maximum at A local minimum will occur 2 units below the midline, at and a local maximum will occur at 2 units above the midline, at Figure shows the graph of the function. Sketch a graph of Determine the midline, amplitude, period, and phase shift.
Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. Draw a graph of Determine the midline, amplitude, period, and phase shift.
Given determine the amplitude, period, phase shift, and horizontal shift. Then graph the function. Begin by comparing the equation to the general form and use the steps outlined in Figure.
Since is negative, the graph of the cosine function has been reflected about the x -axis. Figure shows one cycle of the graph of the function. We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function. A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y -coordinate of the point as a function of the angle of rotation.
Recall that, for a point on a circle of radius r , the y -coordinate of the point is so in this case, we get the equation The constant 3 causes a vertical stretch of the y -values of the function by a factor of 3, which we can see in the graph in Figure.
Notice that the period of the function is still as we travel around the circle, we return to the point for Because the outputs of the graph will now oscillate between and the amplitude of the sine wave is.
What is the amplitude of the function Sketch a graph of this function. A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P , as shown in Figure. Sketch a graph of the height above the ground of the point as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.
Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure. Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.
Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example. Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4.
Putting these transformations together, we find that. A weight is attached to a spring that is then hung from a board, as shown in Figure. As the spring oscillates up and down, the position of the weight relative to the board ranges from in. Assume the position of is given as a sinusoidal function of Sketch a graph of the function, and then find a cosine function that gives the position in terms of. The London Eye is a huge Ferris wheel with a diameter of meters feet. It completes one rotation every 30 minutes.
Riders board from a platform 2 meters above the ground. With a diameter of m, the wheel has a radius of The height will oscillate with amplitude Passengers board 2 m above ground level, so the center of the wheel must be located m above ground level. The midline of the oscillation will be at The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve. Access these online resources for additional instruction and practice with graphs of sine and cosine functions. The sine and cosine functions have the property that for a certain This means that the function values repeat for every units on the x -axis. How does the graph of compare with the graph of Explain how you could horizontally translate the graph of to obtain.
For the equation what constants affect the range of the function and how do they affect the range? Midline of sinusoidal functions from graph. Amplitude of sinusoidal functions from graph.
Midline of sinusoidal functions from equation. Amplitude of sinusoidal functions from equation. Period of sinusoidal functions. Midline, amplitude, and period review Opens a modal.
Period of sinusoidal functions from graph. Period of sinusoidal functions from equation. Graphing sinusoidal functions. Graph sinusoidal functions. Constructing sinusoidal functions. Sinusoidal function from graph Opens a modal. Trig word problem: modeling daily temperature Opens a modal. Trig word problem: modeling annual temperature Opens a modal. Construct sinusoidal functions. Modeling with sinusoidal functions.
The inverse trigonometric functions. Intro to arcsine Opens a modal. Intro to arctangent Opens a modal. Intro to arccosine Opens a modal. Restricting domains of functions to make them invertible Opens a modal. Using inverse trig functions with a calculator Opens a modal.
Inverse trigonometric functions review Opens a modal. Evaluate inverse trig functions. So we have. The functions sine and cosine can be graphed using values from the unit circle, and certain characteristics can be observed in both graphs.
So what do they look like on a graph on a coordinate plane? We can create a table of values and use them to sketch a graph. Again, we can create a table of values and use them to sketch a graph. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. A periodic function is a function with a repeated set of values at regular intervals. The diagram below shows several periods of the sine and cosine functions.
As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function. This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites. Odd symmetry of the sine function: The sine function is odd, meaning it is symmetric about the origin. The graph of the cosine function shows that it is symmetric about the y -axis. This indicates that it is an even function.
The shape of the function can be created by finding the values of the tangent at special angles. However, it is not possible to find the tangent functions for these special angles with the unit circle. We can analyze the graphical behavior of the tangent function by looking at values for some of the special angles. The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined.
At values where the tangent function is undefined, there are discontinuities in its graph. At these values, the graph of the tangent has vertical asymptotes. As with the sine and cosine functions, tangent is a periodic function. This means that its values repeat at regular intervals. If we look at any larger interval, we will see that the characteristics of the graph repeat. The graph of the tangent function is symmetric around the origin, and thus is an odd function.
Calculate values for the trigonometric functions that are the reciprocals of sine, cosine, and tangent. We have discussed three trigonometric functions: sine, cosine, and tangent. Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function. Note that reciprocal functions differ from inverse functions. Inverse functions are a way of working backwards, or determining an angle given a trigonometric ratio; they involve working with the same ratios as the original function.
It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle. It is easy to calculate secant with values in the unit circle. Therefore, the secant function for that angle is. It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle. As with secant, cosecant can be calculated with values in the unit circle. Therefore, the cosecant function for the same angle is.
It can be described as the ratio of the length of the adjacent side to the length of the hypotenuse in a triangle. Cotangent can also be calculated with values in the unit circle. We now recognize six trigonometric functions that can be calculated using values in the unit circle.
Recall that we used values for the sine and cosine functions to calculate the tangent function. We will follow a similar process for the reciprocal functions, referencing the values in the unit circle for our calculations. In other words:. Describe the characteristics of the graphs of the inverse trigonometric functions, noting their domain and range restrictions. Inverse trigonometric functions are used to find angles of a triangle if we are given the lengths of the sides.
Inverse trigonometric functions can be used to determine what angle would yield a specific sine, cosine, or tangent value. Note that the domain of the inverse function is the range of the original function, and vice versa.
However, the sine, cosine, and tangent functions are not one-to-one functions.
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