Introduction
What happens when you divide by zero? You get walls.
Past Knowledge
—invisible walls where the function breaks because the denominator hits zero.
Today's Goal
and cotangent—functions with vertical asymptotes and period .
Future Success
. 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.
Practice Quiz
Practice Quiz
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