Lecture Note (Click To Open) (Video Lecture)

Learning Objectives:

After the completion of this section, you will be able to

1. describe domains, ranges, periods, and vertical asymptotes of the tangent, cotangent, secant, and cosecant functions.

2. graph the tangent, cotangent, secant, and cosecant functions.

Study tip:

You will be able to understand the lecture note easier if you watch the video clips first.
You are to understand and remember that tangent $LaTeX: y=\tan x$ and cotangent functions $LaTeX: y=\cot x$ have a period $LaTeX: \pi$ (NOT $LaTeX: 2\pi$ ).

Key Points:

The period of the tangent function $LaTeX: y=\tan\left(bx\right)$ and cotangent function $LaTeX: y=\cot\:\left(bx\right)$ is $LaTeX: \frac{\pi}{\left|b\right|}$ . (NOT $LaTeX: \frac{2\pi}{\left|b\right|}$ )

The period of the secant function $LaTeX: y=\sec\left(bx\right)$ and cosecant function $LaTeX: y=\csc\left(bx\right)$ is $LaTeX: \frac{2\pi}{\left|b\right|}$ .

You are to remember the standard pattern of the graphs of tangent, cotangent, secanct, and cosecant functions.

tangent graph.png

cotangent graph.png

secant graph.png

cosecant graph.png

Video Resources and Preliminary Exercise Problems:

How to graph the tangent function

Exercise

In the video clip, you learned how to graph the tangent function $LaTeX: y=\tan\:x$ .

The cotangent function is defined by $LaTeX: \frac{x\:coordinate}{y\:coordinate}$ , that is, $LaTeX: \frac{\cos\theta}{\sin\theta}$ . Graph $LaTeX: y=\cot x$ .

Example of Graphing a Tangent Function

Exercise

Graph $LaTeX: y=5\:\tan\left(\frac{1}{3}x\right)$ .

Example of Graphing a Cotangent Function.

Exercise

Graph $LaTeX: y=3\cot\left(\frac{1}{2}x\right)$ .

How to graph the secant graph

Exercise

In the video clip, you learned how to graph $LaTeX: y=\sec x=\frac{1}{\cos x}$ . By following the same process, graph $LaTeX: y=\csc x=\frac{1}{\sin x}$ .