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Mathematics LibreTexts

6.2 Graphs of the Other Trig Functions

In this section, we will explore the graphs of the other four trigonometric functions We’ll begin with the tangent function Recall that in Chapter 5 we defined tangent as y/x or sine/cosine, so you can think of the tangent as the slope of a line through the origin making the given angle with the positive x axis At an angle of 0, the line would be horizontal with a slope of zero As the angle increases towards \(\pi/2\), the slope increases more and more At an angle of \(\pi/2\), the line would be vertical and the slope would be undefined Immediately past \(\pi/2\), the line would have a steep negative slope, giving a large negative tangent value There is a break in the function at \(\pi/2\), where the tangent value jumps from large positive to large negative. 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image245.jpg

We can use these ideas along with the definition of tangent to sketch a graph Since tangent is defined as sine/cosine, we can determine that tangent will be zero when sine is zero:  at -π, 0, π, and so on Likewise, tangent will be undefined when cosine is zero:  at -\(\pi/2\), \(\pi/2\), and so on.

The tangent is positive from 0 to \(\pi/2\) and π to 3\(\pi/2\), corresponding to quadrants 1 and 3 of the unit circle.

Using technology, we can obtain a graph of tangent on a standard grid.

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image247.jpgNotice that the graph appears to repeat itself For any angle on the circle, there is a second angle with the same slope and tangent value halfway around the circle, so the graph repeats itself with a period of π; we can see one continuous cycle from - \(\pi/2\) to \(\pi/2\), before it jumps and repeats itself.                                     

The graph has vertical asymptotes and the tangent is undefined wherever a line at that angle would be vertical: at \(\pi/2\), 3\(\pi/2\), and so on While the domain of the function is limited in this way, the range of the function is all real numbers.

Features of the Graph of Tangent

The graph of the tangent function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image249.gif

  • The period of the tangent function is π
  • The domain of the tangent function is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image251.gif, where k is an integer
  • The range of the tangent function is all real numbers, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image049.gif

With the tangent function, like the sine and cosine functions, horizontal stretches/compressions are distinct from vertical stretches/compressions The horizontal stretch can typically be determined from the period of the graph With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.

Example 1
Find a formula for the function graphed here.

 

The graph has the shape of a tangent function, however the period appears to be 8. We can see one full continuous cycle from -4 to 4, suggesting a horizontal stretch To stretch π to 8, the input values would have to be multiplied byFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image256.gif Since the constant k in File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image258.gifis the reciprocal of the horizontal stretch File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image256.gif, the equation must have form

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image260.gif.

We can also think of this the same way we did with sine and cosine The period of the tangent function is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif but it has been transformed and now it is 8; remember the ratio of the “normal period” to the “new period” is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image263.gifand so this becomes the value on the inside of the function that tells us how it was horizontally stretched.

To find the vertical stretch a, we can use a point on the graph Using the point (2, 2)

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image265.gif.   Since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image267.gif,   a = 2

 

This function would have a formulaFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image269.gif.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image254.jpg

 

Try it Now

  1. Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image271.gif.

 

For the graph of secant, we remember the reciprocal identity where File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image273.gif  Notice that the function is undefined when the cosine is 0, leading to a vertical asymptote in the graph at \(\pi/2\), 3\(\pi/2\), etc Since the cosine is always no more than one in absolute value, the secant, being the reciprocal, will always be no less than one in absolute value Using technology, we can generate the graph The graph of the cosine is shown dashed so you can see the relationship.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image275.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image277.jpg

The graph of cosecant is similar In fact, since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image063.gif, it follows that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image280.gif, suggesting the cosecant graph is a horizontal shift of the secant graph This graph will be undefined where sine is 0 Recall from the unit circle that this occurs at 0, π, 2π, etc The graph of sine is shown dashed along with the graph of the cosecant.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image282.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image284.jpg

 

Features of the Graph of Secant and Cosecant

  1. The secant and cosecant graphs have period 2π like the sine and cosine functions.
  2. Secant has domain File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image251.gif, where k is an integer
  3. Cosecant has domain File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image286.gif, where k is an integer
  4. Both secant and cosecant have range of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image288.gif

 

Example 2

Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image290.gif What is the domain and range of this function?

Solution

The basic cosecant graph has vertical asymptotes at the integer multiples of π Because of the factor File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif inside the cosecant, the graph will be compressed by File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image293.gif, so the vertical asymptotes will be compressed to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image295.gif In other words, the graph will have vertical asymptotes at the integer multiples of 2, and the domain will correspondingly be File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image297.gif, where k is an integer.

The basic sine graph has a range of [-1, 1] The vertical stretch by 2 will stretch this to [-2, 2], and the vertical shift up 1 will shift the range of this function to [-1, 3].

The basic cosecant graph has a range of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image288.gif. The vertical stretch by 2 will stretch this to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image299.gif, and the vertical shift up 1 will shift the range of this function to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image301.gif.

Sketching a graph,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image303.jpg

Notice how the graph of the transformed cosecant relates to the graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image305.gif shown dashed.

 

Try it Now

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image307.jpgGiven the graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image309.gif shown, sketch the graph of  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image311.gif on the same axes.

 

 

 

Finally, we’ll look at the graph of cotangent Based on its definition as the ratio of cosine to sine, it will be undefined when the sine is zero:  at at 0, π, 2π, etc. The resulting graph is similar to that of the tangent In fact, it is a horizontal flip and shift of the tangent function, as we’ll see shortly in Example 3.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image313.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image315.jpg

 

Features of the Graph of Cotangent

  • The cotangent graph has period π
  • Cotangent has domain File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image286.gif, where k is an integer
  • Cotangent has range of all real numbers, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image049.gif

 

In Section 6.1 we determined that the sine function was an odd function and the cosine was an even function by observing the graph and establishing the negative angle identities for cosine and sine Similarly, you may notice from its graph that the tangent function appears to be odd We can verify this using the negative angle identities for sine and cosine:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image318.gif

The secant, like the cosine it is based on, is an even function, while the cosecant, like the sine, is an odd function.

Negative Angle Identities Tangent, Cotangent, Secant and Cosecant

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image320.gif               File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image322.gif

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image324.gif                  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image326.gif

Example 3

Prove that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image328.gif

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image330.gif                                   Using the definition of tangent

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image332.gif                               Using the cofunction identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image334.gif                      Using the definition of cotangent

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image336.gif                       Factoring a negative from the inside

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image338.gif                Using the negative angle identity for cot

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image340.gif

Important Topics of This Section

  • The tangent and cotangent functions
    •           Period
    •           Domain
    •           Range
  • The secant and cosecant functions
    •           Period
    •           Domain
    •           Range
  • Transformations
  • Negative Angle identities

 

Try it Now Answers

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image342.jpg

 

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image344.jpg

Section 6.2 Exercises

Match each trigonometric function with one of the graphs.

1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image346.gif                    2. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image348.gif

3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image350.gif                    4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image352.gif

  IGraphs      IIGraphs

IIIGraphs    IVGraphs

 

Find the period and horizontal shift of each of the following functions.

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image362.gif
6. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image364.gif
7. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image366.gif

8. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image368.gif

9. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image370.gif
10. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image372.gif

11. Sketch a graph of #7 above.

12. Sketch a graph of #8 above.

13. Sketch a graph of #9 above.

14. Sketch a graph of #10 above.

 

15. Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image374.gif.

16. Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image376.gif.

Find a formula for each function graphed below.

17. Graphs  18. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image380.jpg

 

19. http://localhost/filter/graph/svgimg.php?sscr=-5%2C7%2C-8%2C8%2C1%2C1%2C1%2C1%2C1%2C300%2C300%2Cfunc%2C2%2Fsin%28pi%2F4*x%29%2B1%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone  20. http://localhost/filter/graph/svgimg.php?sscr=-4%2C4%2C-8%2C8%2C1%2C1%2C1%2C1%2C1%2C300%2C300%2Cfunc%2C3%2Fcos%28pi*x%29-1%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone

 

  1. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image386.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image388.gif.
  2. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image390.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image388.gif.
  3. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image393.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image395.gif.
  4. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image397.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image395.gif.
  5. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image400.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image402.gif.
  6. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image404.gif, find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image402.gif.

 

Simplify each of the following expressions completely.

27. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image407.gif

28. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image409.gif

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)