Graphs of Other Trigonometric Functions Help

Introudction to Graphs of Other Trigonometric Functions

Because csc x = 1/sin x , the graph of y = csc x has a vertical asymptote everywhere y = sin x has an x -intercept (where sin x = 0). Because sec x = 1/cos x , the graph of y = sec x has a vertical asymptote everywhere y = cos x has an x -intercept. The period for y = csc x and y = sec x is 2 π . The graph for y = csc x is shown in Figure 13.26, and the graph for y = sec x is shown in Figure 13.27.

Fig. 13.26

Fig. 13.27

The domain for y = csc x is all real numbers except for the zeros of sin x , x ≠ ..., −2 π , − π , 0, π , 2 π , .... The range is (−∞, −1] ∪ [1, ∞). The domain for y = sec x is all real numbers except for the zeros of cos x , x ≠ ..., −3 π /2, − π /2, π /2, 3 π /2, .... The range is (−∞, −1] ∪ [1, ∞). Because y = sin x is an odd function, y = csc x is also an odd function. Because y = cos x is an even function, y = sec x is also an even function.

We can sketch the graphs of y = csc x and y = sec x using the graphs of y = sin x and y = cos x. We will sketch the vertical asymptotes as well as the graphs of y = sin x or y = cos x using dashed graphs.

The graph of y = sin x is given in Figure 13.28. Vertical asymptotes are sketched for every x -intercept.

Fig. 13.28

The vertex for each piece on the graph of y = csc x is also a vertex for y = sin x.

Fig. 13.29

Then we can plot a point to the left and right of each vertex (staying inside the vertical asymptotes) to show how fast the graph gets close to the vertical asymptotes.

Table 13.5

Fig. 13.30

Now we can draw ∪ or ∩ through the points.

Fig. 13.31

These steps also work for the graph of y = sec x .

The period for the functions y = tan x and y = cot x is π instead of 2 π as it is with the other trigonometric functions. These graphs also have vertical asymptotes. The graph of y = tan x (= sin x /cos x ) has a vertical asymptote at each zero of y = cos x . The graph of y = cot x (= cos x / sin x ) has a vertical asymptote at each zero of y = sin x. The graph of y = tan x is shown in Figure 13.32, and the graph of y = cot x is shown in Figure 13.33.

Fig. 13.32

Fig. 13.33

The domain of y = tan x is all real numbers except the zeros of y = cos x , x ≠ ..., −3 π /2, − π /2, π /2, 3 π /2, .... The domain for y = cot x is all real numbers except for the zeros of y = sin x , x ≠ ..., −2 π , − π , 0, π , 2 π , .... The range for both y = tan x and y = cot x is all real numbers. Both are odd functions.

The transformations of these are similar to those of the other trigonometric functions. For functions of the form y = a csc k ( x − b ) and y = a sec k ( x − b ), the period is 2 π / k , and the phase shift is b . For functions of the form y = a tan k ( x − b ) and y = a cot k ( x − b ), the period is π/k , and the phase shift is b . The term amplitude only applies to the sine and cosine functions.

Practice problems for this concept can be found at: Trigonometry Practice Test.