# Precalculus

## Trigonometry # 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.