Graphs of Secant and Cosecant Functions

Definitions of Secant and Cosecant

The secant function is defined as the reciprocal of the cosine:

\[\sec t = \frac{1}{{\cos t}}.\]

Similarly, the cosecant function is the reciprocal of the sine:

\[\csc t = \frac{1}{{\sin t}}.\]

The Graph and Properties of the Secant Function

Domain and Codomain of the Secant Function

Since \(\cos t = 0\) at \(t = \frac{\pi }{2} + \pi n,\,n \in \mathbb{Z},\) then the function \(\sec t\) is not defined at these points. At each of these points, the secant function has vertical asymptotes.

It is known that the cosine function varies between \(-1\) and \(1.\) The secant function being the reciprocal of the cosine takes the values on the set \(\left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right).\)

Hence, we can write:

\[\text{dom}\left( {\sec t} \right) = \left\{ {t \in \mathbb{R} \left|\, t \ne \frac{\pi }{2} + \pi n\right., n \in \mathbb{Z}} \right\},\]

\[\text{codom}\left( {\sec t} \right) = \left\{ {y \in \mathbb{R} \left|\,\left| y \right| \ge 1\right. } \right\}.\]

Parity of the Secant Function

The function \(\sec t\) is even:

\[\sec \left( { - t} \right) = \frac{1}{{\cos \left( { - t} \right)}} = \frac{1}{{\cos t}} = \sec t.\]

Periodicity of the Secant Function

Like the cosine, the secant function is periodic with the least period \(2\pi:\)

\[\sec \left( {t + 2\pi n} \right) = \sec t,\]

where \(n \in \mathbb{Z}.\)

Local Maxima and Minima of the Secant Function

The function \(\sec t\) has local minima \(y_{\min} = 1\) at the points \(t = 2\pi n,\,n \in \mathbb{Z}\) where \(\cos t = 1.\) It also has local maxima \(y_{\max} = -1\) at \(t = \pi + 2\pi n,\,n \in \mathbb{Z}\) where \(\cos t = -1.\)

Note that the local maximum \(y_{\max} = -1\) is less than the local minimum \(y_{\min} = 1.\) This is because the secant function is discontinuous and the points of local maximum and minimum lie on different branches of the graph.

Graph of the Secant Function \(y = \sec t\)

The graph of the secant function consists of an infinite number of alternating convex and concave curves separated by vertical asymptotes.

The Graph and Properties of the Cosecant Function

Domain and Codomain of the Cosecant Function

The function \(\csc t = \frac{1}{{\sin t}}\) is undefined at the points \(t = \pi n,\,n \in \mathbb{Z}\) where the denominator is equal to zero. At these points, the cosecant function has vertical asymptotes.

The range of the cosecant function is the same as for secant: \(\left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right).\)

So, using set builder notation, we have

\[\text{dom}\left( {\csc t} \right) = \left\{ {t \in \mathbb{R} \left|\, t \ne \pi n\right., n \in \mathbb{Z}} \right\},\]

\[\text{codom}\left( {\csc t} \right) = \left\{ {y \in \mathbb{R} \left|\,\left| y \right| \ge 1\right. } \right\}.\]

Parity of the Cosecant Function

Since the sine function is odd, the cosecant function is also odd:

\[\csc \left( { - t} \right) = \frac{1}{{\sin \left( { - t} \right)}} = \frac{1}{{ - \sin t}} = - \frac{1}{{\sin t}} = - \csc t.\]

Periodicity of the Cosecant Function

Like the sine function, the cosecant function is periodic with the least period \(2\pi:\)

\[\csc \left( {t + 2\pi n} \right) = \csc t,\]

where \(n \in \mathbb{Z}.\)

Local Maxima and Minima of the Cosecant Function

The cosecant function has local minima \(y_{\min} = 1\) at the points \(t = \frac{\pi }{2} + 2\pi n,\,n \in \mathbb{Z}\) in which \(\sin t = 1,\) and local maxima \(y_{\max} = -1\) at the points \(t = \frac{3\pi }{2} + 2\pi n,\,n \in \mathbb{Z}\) in which \(\sin t = -1.\)

Graph of the Cosecant Function \(y = \csc t\)

Given that the cosine function is even and using the cofunction identity, we can represent the cotangent function in the form:

\[\csc t = \frac{1}{{\sin t}} = \frac{1}{{\cos \left( {\frac{\pi }{2} - t} \right)}} = \frac{1}{{\cos \left( {t - \frac{\pi }{2}} \right)}} = \frac{1}{{\cos \left( {t - \frac{\pi }{2}} \right)}} = \sec \left( {t - \frac{\pi }{2}} \right).\]

This means that the cosecant graph is the same as the secant graph, but only shifted by \({\frac{\pi }{2}}\) to the right.

See solved problems on Page 2.

Precalculus

Trigonometry