4. Graphs of tan, cot, sec and csc

t = θ = 0

y = 100 tan(0) = 0

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(For more on periodic functions and to see `y = tan x` using degrees, rather than radians, see Trigonometric Functions of Any Angle.)

t = θ = 0

y = 100 cot(0) = `oo`

Frame rate = 0 fps

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The Graph of y = sec x

We could laboriously draw up a table with millions of values, or we could work smart and recall that

`sec x=1/(cos x)`

We know the sketch for y = cos x and we can easily derive the sketch for y = sec x, by finding the reciprocal of each y-value. (That is, finding `1/y` for each value of y on the curve `y = cos x`.)

For example (angles are in radians):

`x` 0 1 1.55 2 3 4
`y = cos x` 1 0.54 0.02 -0.42 -0.99 -0.65
`1/y = sec x` 1 1.85 48.09 -2.4 -1.01 -1.53

I included a value just less than `π/2=1.57` so that we could get an idea of what goes on there. When `cos x` is very small, `sec x` will be very large.

After applying this concept throughout the range of x-values, we can proceed to sketch the graph of `y = sec x`.

First, we graph `y = cos x` and then `y = sec x` immediately below it. Compare the y-values in each of the 2 graphs and assure yourself they are the reciprocal of each other.

y = cos x

Graph of cos x

y = sec x

Graph of sec x

We draw vertical asymptotes at the values where `y = sec x` is not defined. That is, when `x = ..., -(5π)/2, -(3π)/2, -π/2, π/2, (3π)/2, (5π)/2, ...`

You will notice that these are the same asymptotes that we drew for `y = tan x`, which is not surprising, because they both have `cos x` on the bottom of the fraction.

Interactive Secant curve

As before, run the animation and observe the features of the graph. Then change the radius of the circle (and run the animation again) to see the effect of different energy levels.

You can see the graph of the cosine curve (in light color) as a guide.

t = θ = 0

y = 100 sec(0) = 0

Frame rate = 0 fps

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y = csc x


Try to sketch it before checking your answer with this cosecant graph interactive.

You can see the graph of the sine curve (in light color) as a guide.

t = θ = 0

y = 100 csc(0) = 0

Frame rate = 0 fps

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You may also be interested in:

The next section in this chapter shows some Applications of Trigonometric Graphs.

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