The Graph of Tangent
Simple Explanation
y = tan x looks very different from sine and cosine β it has vertical asymptotes wherever cos x=0 (at x=Ο/2+kΟ), is unbounded (covers all real numbers between each pair of asymptotes), and repeats every Ο instead of 2Ο.
Why Do We Need It?
Understanding tangent's asymptotic behavior is essential, since it behaves fundamentally differently from the bounded, smooth sine and cosine curves.
See It
A repeating curve with vertical asymptotes, shooting up toward positive infinity and down toward negative infinity near each asymptote
Formula
Properties of y = tan x
Domain: all x except Ο/2+kΟ. Range: all real numbers. Period: Ο
y=tanx has vertical asymptotes wherever cosx=0, is unbounded (covers all real numbers), and repeats every Ο β half the period of sine and cosine.
- k
- β any integer, giving the full set of excluded x-values
When to use it: Whenever identifying or sketching the tangent graph, especially its asymptotes.
Worked Example
Evaluate tangent and explain its asymptotic behavior
Find tan(Ο/4), and explain what happens to the graph near x=Ο/2.
Why Does This Work?
tan(x) is literally defined as sin(x)/cos(x) β wherever the denominator cos(x) hits exactly zero, the ratio is undefined, and the graph shoots off toward positive or negative infinity on either side.
Real-Life Example
Modeling the length of a shadow as the sun sets
The length of a shadow cast by a vertical pole depends on the tangent of the sun's angle above the horizon.
As the sun approaches the horizon (angle approaching 0Β° from a different reference), the tangent-based shadow length can grow without bound β a real-world instance of tangent's unbounded behavior.
Practice
Find tan(0).
MediumCommon mistake
Assuming tangent has the same 2Ο period as sine and cosine β tangent actually repeats every Ο, half as often.
Quick Review
- y=tanx: undefined at Ο/2+kΟ, range all reals, period Ο.
- tan(x) = sin(x)/cos(x); asymptotes occur where cos(x)=0.
- Unlike sine and cosine, tangent is unbounded.