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.
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.
The vertex for each piece on the graph of y = csc x is also a vertex for y = sin x.
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.
x |
csc x |
−1.8π |
1.7 |
−1.2π |
1.7 |
−0.8π |
−1.7 |
−0.2π |
−1.7 |
0.2π |
1.7 |
0.8π |
1.7 |
1.2π |
−1.7 |
1.8π |
−1.7 |
Now we can draw ∪ or ∩ through the points.
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.
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.
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