Lesson 4.9

Graphing Cotangent

Cotangent is the reciprocal of tangent — . Its graph is a decreasing S-curve with asymptotes where sine equals zero.

Introduction

Past Knowledge

You graphed with period and asymptotes at .

Today's Goal

Graph and compare its features to tangent.

Future Success

Cotangent helps you understand cosecant, which shares the same asymptotes.

Key Concepts

Key Features of

Feature	Value
Period
Asymptotes	(where )
Zeros	(where )
Behavior	Always decreasing (opposite of tan)

Purple = cot(x) · Red dashed = asymptotes · Green = zeros

Tangent vs. Cotangent — Quick Comparison

Both have period . Tangent increases with asymptotes where cos = 0. Cotangent decreases with asymptotes where sin = 0. Their zeros and asymptotes are swapped!

Worked Examples

Basic

Cotangent Asymptotes

Question: Where are the asymptotes of in ?

Step 1: Asymptotes where : at .

Final Answer:

Intermediate

Value of Cotangent

Question: Find .

Step 1:

Final Answer:

Advanced

Modified Cotangent

Question: Find the period and asymptotes of in .

Step 1: Period .

Step 2: Asymptotes:

Step 3: In :

Final Answer: Period . Three complete branches in .

Common Pitfalls

Swapping Tangent and Cotangent Asymptotes

Tangent: asymptotes where . Cotangent: asymptotes where . They're reciprocals, so their asymptotes and zeros are swapped.

Real-Life Applications

Sundial Design

The position of a shadow on a sundial involves the cotangent function. As the sun's elevation angle changes, the shadow length is proportional to . At sunrise and sunset (), the shadow stretches toward infinity — matching the asymptotic behavior of the cotangent graph.