Trigonometry; Graphs of sec θ, cosec θ, and cot θ

This section explores graphs of secant θ, cosecant θ, and contagent θ. This is a continuation of trigonometry previous article.

Sec θ Graph

First let us look at the graph of the y = sec θ function. The following is an important property of the sec θ graph.

The graph of y = sec θ has symmetry in the y-axis and repeats itself every 360°. It has vertical asymptotes at all values of θ for which cos θ = 0, for instance at θ = 90° + 180n°, n ∈ ℤ the interval of -180° ≤ θ ≤ 180°.

Below is a typical graph of the y = sec θ graph. The graph has been sketched in t
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Notice that for each value of θ, the value of sec θ is the reciprocal of the corresponding value of cos θ. Also notice that as cos 0° = 1 ⇒ sec 0° = 1, as cos 180° = -1 ⇒ sec 180° = -1. The graph of y = sec θ has been redrawn below with the y = cos θ curve in green. This should help observe the relationship of the sec θ curve with the cos θ curve.
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You will notice that;

• As θ approaches 90° from left, cos θ is +ve but approaches zero and so sec θ is +ve but becoming increasing large.
• As θ approaches 90° from the right, cos θ is –ve but approaches zero, and so sec θ is –ve but becoming increasingly large negative.
• At θ = 90° there is a break in the sec θ at θ = 90° (you may come across this written as ± ∞ for this point or value)

Graph of y = cosec θ

Next we shall explore the graph of cosec θ. You must know that;
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It also follows that;
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So we first draw the sketch of y = sec θ as shown below;
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sec (θ – 90)° is a translation to the right by 90°. Therefore we translate the graph of y=sec θ by 90° to the right as shown below; The following is the graph of y=cosec θ;
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Below is an important property to remember about the y=cosec θ graph.

The graph of y=cosec θ , θ ∈ ℝ (θ is an element of the set of real numbers), has vertical asymptotes at all the values of θ for which sin θ = 0, ie at θ = 180n°, n ∈ ℤ and the curve repeats itself every 360°

Graph of y = cot θ

This is also a very easy graph to draw. It is also similar to the y = tan θ curve. The sketch of the graph y = cot θ has been drawn below;

The graph of y = cot θ, θ ∈ ℝ (θ is an element of the set of real numbers) has a vertical asymptotes at all values of θ for which sin θ = 0, ie at θ = 180n°, n ∈ ℤ, and the curve repeats itself every 180°.

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Properties of the y=cot θ curve are;

• At the values of θ where asymptotes occur on y = tan θ , the graph of y = cot θ passes through the θ – axis.
• At the values of θ where y = tan θ crosses the θ – axis, y=cot θ has asymptotes.
• When tan θ is small and positive, cot θ is large and positive; when tan θ is large and positive cot θ is small and positive. Similarly for negative values as well.

Notice you can continue further to apply translations to the graph functions.