Lesson 10.4

Graphs of Tangent and Cotangent

What happens when you divide by zero? You get walls.

Introduction

Sine and Cosine are polite functions: they wave smoothly and never cause trouble. Tangent and Cotangent are different. Because they involve division, they have Asymptotes—invisible walls where the function breaks because the denominator hits zero.

Prerequisite Connection

You know that and . Division by zero creates asymptotes.

Today's Increment

We graph tangent and cotangent—functions with vertical asymptotes and period .

Why This Matters

In Calculus, Tangent is key for integration techniques. The derivative of is , connecting to secant-based integrals.

The Tangent Graph

Since , tangent explodes whenever . Where is cosine zero? At

Key Feature: Period is

Unlike sine/cosine ($2\pi$), tangent repeats every .

The Cotangent Graph

Cotangent is . It explodes when Sine is zero (at ). It looks like tangent, but flipped and shifted.

Worked Examples

Example 1: Finding Asymptotes

Find the vertical asymptotes of .

Set Argument to Asymptote

Normal tangent has asymptotes at .

So we set the inside part .

Solve for x

Divide everything by 2.

Example 2: Period and Phase Shift

Find the period and phase shift of .

Factor out B

Calculate Period

Period for tan is .

Identify Phase Shift

Phase shift is Right .

Example 3: Graph from Equation (Advanced)

Sketch one period of .

Identify Parameters

(reflected), , Phase Shift = right.

Find Asymptotes

Normal cot asymptotes at . With shift: .

Graph Properties

The negative sign reflects the graph—instead of decreasing, it now increases from left to right within each period.