# Secant, Cosecant and Cotangent Graphs

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A series of free, online Trigonometry Video Lessons.
Videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

• the asymptotes of secant, cosecant and cotangent
• the cotangent graph
• transformation of the cotangent graph
• the secant and cosecant graphs
• transformation of the secant and cosecant graphs

### Asymptotes of Secant, Cosecant, and Cotangent

To find the x-intercepts and asymptotes of secant, cosecant, and cotangent, rewrite them in terms of sine and cosine. Notice that since secant and cosecant have 1 in the numerator and a trig function in the denominator, they can never equal zero; they do not have x-intercepts. The vertical asymptotes of the three functions are whenever the denominators are zero.

### Cotangent Graph

Cotangent is the reciprocal trig function of tangent function and can be defined as cot θ = cos θ /sin θ. It is an odd function, meaning cot(− θ) = − cot(θ), and it has the property that cot(θ + π) = cot(θ). Because sine is the denominator, and the function is undefined when sin(θ) = 0, the cotangent graph has vertical asymptotes at all integer multiples of π, when sin(θ) = 0.

In this video, we determine the graph of cotangent and determine the key properties of the cotangent graph. Illustrates the graph of the cotangent function using the cotangent segment. Explains how to graph cotangent using reciprocal values of the tangent function.

How to graph the tangent and cotangent graph

### Transforming the Cotangent Graph

Graphing Tangent and Cotangent over Different Periods
Graph y = Atan(Bx) and y = Acot(Bx)

This video provides an example of graphing the cotangent function with a different period and a vertical stretch.
y = 2cot(x/4)

### Secant and Cosecant Graphs

This video explains how you can use cosine function and the reciprocal identity to graph the secant function.

This video explains how you can use the sine function and the reciprocal identity to graph the cosecant function.

How to graph Cosecant and Secant using Sine and Cosine.
Determine the domain and range of the two functions. Determine the period of the two functions.

### Transforming the Secant and Cosecant Graphs

To graph secant and cosecant, find values of the reciprocal functions and plot them on the coordinate plane. Unlike the graphs of sine and cosine, secant and cosecant have vertical asymptotes whenever the cosine and sine equal zero, respectively. Graphing transformations is made easier by substituting theta for the quantity in parenthesis and solving for x. Also, notice that neither graph has x-intercepts.

How to graph the secant function that has an amplitude and vertical transformation
y = 3 + 2sec x

This video provides an example of describing and graphing a transformation of the cosecant function.
y = 2csc(2πx + π) + 3

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