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Trigonometric Functions and Their Graphs:
  The Co-Functions 
(page 3 of 3)

Sections: The sine and cosine, The tangent, The co-functions

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What about the co-functions, the secant, the cosecant, and the cotangent?

The cosecant is the reciprocal of the sine. Wherever the sine is zero, the cosecant will be undefined, so there will be a vertical asymptote. Wherever the sine reaches its maximum value of 1, the cosecant will reach its minimum value of 1; wherever the sine reaches its minimum value of –1, the cosecant will reach its maximum value of –1. Wherever the sine is positive but less than 1, the cosecant will be positive but greater than 1; wherever the sine is negative but greater than –1, the cosecant will be negative but less than –1.




So I'll lightly draw the sine wave...

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graph with gray sine wave from -pi/2 to 5pi/2





...I'll draw vertical asymptotes through its zeroes and note the min/max points...


same graph, with dots at (-pi/2, -1), (pi/2, 1), (3pi/2, -1), and (5pi/2, 1) and asymptotes at x = 0, pi, 2pi





...and then I'll fill in the graph.


graph with cosecant curve added

The Cosecant Graph

By using the same reasoning with the cosine wave, I can create the secant graph:

graph of secant, showing cosine wave in gray for comparison

The Secant Graph

The secant and cosecant have periods of length , and we don't consider amplitude for these curves.

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The cotangent is the reciprocal of the tangent. Wherever the tangent is zero, the cotangent will have a vertical asymptote; wherever the tangent has a vertical asymptote, the cotangent will have a zero. And the signs on each interval will be the same. So the cotangent graph looks like this:

graph of cotangent, with tangent shown in gray for comparison

The Cotangent Graph

The cotangent has a period of π, and we don't bother with the amplitude.

When you need to do the graphs, you may be tempted to try to compute a lot of plot points. But all you really need to know is where the graph is zero, where it's equal to 1, and / or where it has a vertical asymptote. If you know the behavior of the function at zero, π/2, π, 3π/2, and , then you can fill in the rest. That's really all you "need".

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Cite this article as:

Stapel, Elizabeth. "Trigonometric Functions and Their Graphs: Co-functions." Purplemath. Available from Accessed



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