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Trigonometric Functions

Graphing Sine and Cosine Functions

Sine and cosine can be interpreted in multiple ways, such as the relationship between angles and side lengths in triangles, and as the coordinates of points on the unit circle. Since sine and cosine return exactly one output value for every input angle, they can also be interpreted as functions.

Sine and cosine can be evaluated for any value of Thus, their domain is all real numbers. Every output, however, ranges from to — the range of both functions is


The Graphs of and

The parent functions of sine and cosine functions are and respectively. Thus, it is important to know how their graphs look and some of their characteristics. Graphing them is done by first evaluating some inputs and obtaining their corresponding outputs using a table. Here, radians will be used, but the functions can be evaluated for degrees as well.

Plotting these points and connecting them with smooth curves gives the graphs.

Both functions have the amplitude which is half the difference between the greatest and least function values. These functions are both periodic, as they have a repeating pattern. The smallest repeating part of the graph is called a cycle. The horizontal length of the cycles is called the function's period, which is for both and


Stretching and Shrinking Sine and Cosine Functions

To find more sine and cosine functions, the parent functions can be stretched or shrunk both horizontally and vertically, leading to the functions The amplitude is affected by the factor Notice though that an amplitude is always positive, while could be negative. Similarly, the period is affected by the constant A larger shrinks the functions horizontally, leading to a shorter period. Here, as well, could be negative while a period is always positive. Thus, the amplitude and period of these functions are as follows.

Amplitude = Period =


The graph shows the function

Using the graph, find the amplitude and period of the function. Then, verify them using the rule.

Show Solution

We can find the amplitude of the function by reading the greatest and least function values and calculating half the difference between them. From the graph, it can be seen that the greatest value is and the least is

Calculating the difference between these values and dividing by two gives us The period is found by identifying the horizontal length of a cycle. We'll look at the cycle starting at the origin, ending the next time the graph passes the -axis from below.

This cycle starts at and ends at Thus, we can conclude that When a sine function is written in the form which the given function is, the amplitude is the absolute value of Therefore, we find that For the period, we divide by the absolute value of Thus, the previous results have now been verified.


Translating Sine and Cosine Functions

By translating the functions and every sine and cosine function can be created. The resulting function rules are then

where is the horizontal and is the vertical translation. The horizontal line is called the midline of the graph and is located in-between the -values of the maximums and minimums. The graph's intersections with the midline corresponds to the -intercepts of the parent function.


Graphing Sine and Cosine Functions

Viewing sine and cosine functions as transformations of their parent functions, and makes it easier to graph these by hand. As an example, consider the function


Find the amplitude, period, and translation of the function

Identifying the amplitude, period, and translation allows the function to be viewed as the transformation that it is. For this function, the amplitude is the period is and it's translated unit upward.


Draw the midline

As the points intersecting the midline correspond to the -intercepts of the parent function, the midline should be the first thing graphed. Here, the midline is


Plot some key points on the graph

Key points ranging over at least one cycle of the function should now be plotted, as they can then be connected with a smooth curve to create the graph. These key points are the maximums, minimums, and intersections with the midline. Starting with the maximums, they occur at for which is once every cycle. The period of is and it's not been translated horizontally. Thus, the maximums of occur at As the midline is and the amplitude is the maximums will all have the -values They can now be plotted in the graph with the midline.

The minimums are located horizontally between the maximums, which is at for These have the -values

Lastly, in-between every neighboring maximum and minimum are the intersections with the midline: These points have the -value


Draw the graph

The graph can now be drawn through the plotted points, continuing periodically in both directions.


Riding a certain ferris wheel can be described using the function where is the height above ground in feet, and is the time since the ride started given in minutes.

Using the function rule, answer the following questions.

  • What is the diameter of the ferris wheel?
  • One ride on the ferris wheel is two full rotations. How long does a ride take?
  • How high above ground is the entry of the ferris wheel?
Show Solution


Diameter of the ferris wheel

The diameter of the ferris wheel can be thought of as the difference in height between the highest and lowest point. Note that the amplitude is half this difference. Thus, the diameter is equal to twice the amplitude of the function. From the rule we find that the amplitude is which gives us as the height is given in feet.


Length of a ride

One full rotation of the wheel corresponds to one cycle of the function, as it will then repeat itself. Thus, one full rotation takes as long as one period of the function is. We can find the period as usually: One ride is twice this time, minutes.


Entry's height above ground

Realistically, the ride should start at the same height as the entry. That is, when Therefore, substituting into the function gives us the height of the entry.

As the angle is equal to it corresponds to the coordinate on the unit circle. Thus, is equal to

We can now conclude that the entry to the ferris wheel is feet off ground level.

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