While sine and cosine trace gentle waves, tangent and cotangent produce a very different pattern: repeating curves separated by vertical asymptotes. In this lesson you will learn how to sketch these graphs, identify their asymptotes, and understand why their period is only $\pi$ instead of $2\pi$.
Today's hook, The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. As $x$ gets closer to $90^\circ$ from below, $\cos x$ gets closer to 0 while $\sin x$ stays close to 1. What do you think happens to the value of $\tan x$? And what does this mean for the graph near $x = 90^\circ$?
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. As $x$ gets closer to $90^\circ$ from below, $\cos x$ gets closer to 0 while $\sin x$ stays close to 1. What do you think happens to the value of $\tan x$? And what does this mean for the graph of $y = \tan x$ near $x = 90^\circ$?
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The two moves
Work through the core explanation before applying it.
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The two moves
+5 XP to read
Tangent and cotangent have the same period and the same range, but their asymptotes and intercepts are in completely different places. Master these two patterns and you can sketch any transformed version.
Every tan/cot problem comes down to two core facts: tangent is undefined where cosine is zero, and cotangent is undefined where sine is zero. Both have period $\pi$ and range all real $y$.
$\pi$
$y = \tan x$
Period: $\pi$. Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$. Passes through $(0, 0)$. Range: all real $y$.
$y = \cot x$
Period: $\pi$. Vertical asymptotes: $x = n\pi$. Passes through $(\frac{\pi}{2}, 0)$. Range: all real $y$.
Key insight
Tan and cot have period $\pi$ (not $2\pi$). This is because $\tan(x + \pi) = \tan x$.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
The shape and key features of $y = \tan x$ and $y = \cot x$
The period of tangent and cotangent is $\pi$
The locations of vertical asymptotes for both functions
Understand
Concepts
Why tangent has vertical asymptotes where cosine is zero
Why the period of tangent is $\pi$ instead of $2\pi$
How cotangent relates to tangent by reflection and shift
Can do
Skills
Sketch the graphs of $y = \tan x$ and $y = \cot x$
Identify asymptotes, intercepts, and period from an equation
Sketch transformed tangent and cotangent graphs
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Key terms
Tangent CurveThe graph of $y = \tan x$; has vertical asymptotes and period $\pi$.
Cotangent CurveThe graph of $y = \cot x$; has vertical asymptotes and period $\pi$.
AsymptoteA line that a curve approaches but never touches.
Period ($\pi$)The horizontal length of one complete cycle for tan and cot.
UndefinedA function value that does not exist, e.g. $\tan(\pi/2)$.
Vertical AsymptoteA vertical line $x = a$ where a function grows without bound.
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Graph of $y = \tan x$
Work through the core explanation before applying it.
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Graph of $y = \tan x$
core concept · +3 XP at end
Because $\tan x = \frac{\sin x}{\cos x}$, the tangent function is undefined wherever $\cos x = 0$. This occurs at:
$$x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$$
At these values, the graph has vertical asymptotes. Between each pair of asymptotes, the tangent graph forms a smooth, increasing curve that passes through the $x$-axis.
$y = \tan x$ has vertical asymptotes (red dashed) at $x = \frac{\pi}{2} + n\pi$ and period $\pi$
Key Features of $y = \tan x$
Period: $\pi$ (repeats every $\pi$ radians)
Domain: All real $x$ except $x = \frac{\pi}{2} + n\pi$
Range: All real $y$
$x$-intercepts: $x = n\pi$ (where $\sin x = 0$)
Asymptotes: $x = \frac{\pi}{2} + n\pi$
Behaviour: Always increasing between asymptotes
Why the period is $\pi$. The tangent function has period $\pi$ because both sine and cosine change sign when we add $\pi$, so their ratio stays the same: $\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \tan x$.
$y = \tan x$ is undefined wherever $\cos x = 0$, i.e. at $x = \frac{\pi}{2} + n\pi$; Period = $\pi$ (not $2\pi$, a very common exam trap)
Pause, copy the $y = \tan x$ key facts: undefined at $x = \frac{\pi}{2} + n\pi$ (asymptotes), period $= \pi$ (half a sine cycle, this is the classic exam trap), and range = all reals into your book.
Did you get this? True or false: the period of $y = \tan x$ is $2\pi$, the same as $y = \sin x$.
Quick check: At which $x$-values does $y = \tan x$ have vertical asymptotes?
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Graph of $y = \cot x$
core concept
We just saw that $\tan x = \frac{\sin x}{\cos x}$ is undefined where $\cos x = 0$, giving asymptotes at $x = \frac{\pi}{2} + n\pi$. That raises a question: what does the reciprocal $\cot x = \frac{\cos x}{\sin x}$ look like, and where are its asymptotes? This card answers it → $\cot x$ is undefined where $\sin x = 0$, so asymptotes shift to $x = n\pi$ (where $\tan x$ had zeros).
Beyond the syllabus. The tangent graph is Core. MAV-11-04 asks for the graph of $y = \tan x$ with its period and its asymptotes, and you will be examined on it. The cotangent graph is not named anywhere in the Year 11 syllabus. It is here because it makes the asymptote rule general, tangent undefined where $\cos x = 0$ and cotangent undefined where $\sin x = 0$, and that pattern is worth one read. What you need for the exam is the tangent graph, its period of $\pi$ and its asymptotes at $x = \frac{\pi}{2} + n\pi$, so put your practice there rather than into sketching transformed cotangent curves.
Because $\cot x = \frac{\cos x}{\sin x}$, the cotangent function is undefined wherever $\sin x = 0$. This occurs at:
Note that $y = \cot x$ is related to $y = \tan x$ by a reflection and shift. Specifically:
$$\cot x = \tan\left(\frac{\pi}{2} - x\right)$$
$y = \cot x$ is undefined wherever $\sin x = 0$, i.e. at $x = n\pi$; Asymptotes at $x = n\pi$, these are where tan crosses zero, which is easy to mix up
Pause, copy the $\cot x$ key: undefined at $x = n\pi$ (asymptotes here = zero-crossings of $\tan x$), this role-reversal is the easy mix-up to watch for into your book.
Quick check: Where are the asymptotes of $y = \cot x$?
Worked examples · 3 in a row, reveal as you go
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Worked Example, Sketching $y = \tan x$
+5 XP for trying first
Sketch $y = \tan x$ for $-\frac{\pi}{2} < x < \frac{3\pi}{2}$ and label the asymptotes and $x$-intercepts.
Your turn first. Try it yourself before viewing the solution.
Answer: Period = $\frac{\pi}{2}$, Asymptotes: $x = \frac{\pi}{4} + \frac{n\pi}{2}$
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Worked Example, Sketching $y = \cot x$
+5 XP for trying first
Sketch $y = \cot x$ for $0 < x < 2\pi$ and label the asymptotes and intercepts.
Your turn first. Try it yourself before viewing the solution.
SOLUTION
3 marks · HSC band 4+
Step 1: Find the asymptotes.
$$x = 0, \; \pi, \; 2\pi$$
Step 2: Find the $x$-intercepts.
$$x = \frac{\pi}{2}, \; \frac{3\pi}{2}$$
Step 3: Draw smooth decreasing branches between each pair of asymptotes, passing through the intercepts.
Answer: Asymptotes at $x = 0, \pi, 2\pi$; intercepts at $\left(\frac{\pi}{2}, 0\right)$ and $\left(\frac{3\pi}{2}, 0\right)$
Common traps, the 3 that cost marks
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Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Using period $2\pi$ for tangent or cotangent
Students sometimes apply the sine/cosine period formula $\frac{2\pi}{b}$ to tangent. The correct period for tangent is $\frac{\pi}{b}$. Tangent and cotangent have period $\pi$, not $2\pi$.
Trap 02
Drawing curves that touch or cross the asymptotes
The branches of tangent and cotangent approach the asymptotes but never touch them. Asymptotes are drawn as dashed lines that the curve approaches but does not cross.
Trap 03
Confusing the asymptotes of tangent and cotangent
Tangent has asymptotes where $\cos x = 0$ (odd multiples of $\frac{\pi}{2}$), while cotangent has asymptotes where $\sin x = 0$ (multiples of $\pi$). Fix: Tangent asymptotes: $\frac{\pi}{2} + n\pi$. Cotangent asymptotes: $n\pi$.
Quick-fire practice · 5 reps +2 XP per reveal
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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$y = \tan x$ for $-\pi < x < \pi$, describe the asymptotes and intercepts.
State the period and asymptotes of $y = \cot(3x)$.
Show answer
Period: $\frac{\pi}{3}$. Asymptotes: $3x = n\pi \Rightarrow x = \frac{n\pi}{3}$.
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Explain why $\tan(x + \pi) = \tan x$ for all values where $\tan$ is defined.
Show answer
$\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \tan x$. Since the function repeats every $\pi$, the period is $\pi$.
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Revisit, tangent near $90^\circ$
+5 XP for checking
As $x \to 90^\circ$ from below, $\cos x \to 0^+$ and $\sin x \to 1$, so $\tan x = \frac{\sin x}{\cos x} \to +\infty$. This means the graph of $y = \tan x$ has a vertical asymptote at $x = \frac{\pi}{2}$. The curve rises steeply and never crosses this line.
Return to your original answer from Section 01. What did you get right? What has changed in your thinking?
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 4
Sketch $y = \tan x$
Sketch $y = \tan x$ for $-\frac{\pi}{2} < x < \frac{3\pi}{2}$. Label all asymptotes and $x$-intercepts. 3 MARKS
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View comprehensive answer
Asymptotes at $x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}$ [1].
$x$-intercepts at $x = 0, \pi$ [1].
Smooth increasing branches between asymptotes [1].
ApplyBand 4
Period and asymptotes of $y = \cot(2x)$
Find the period and the equations of the vertical asymptotes of $y = \cot(2x)$. 3 MARKS
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View comprehensive answer
Period = $\frac{\pi}{2}$ [1].
$2x = n\pi \Rightarrow x = \frac{n\pi}{2}$ [2].
AnalyseBand 5
Why the period of tangent is $\pi$
Explain why $\tan(x + \pi) = \tan x$ for all values of $x$ where $\tan x$ is defined. Use this result to explain why the period of $y = \tan x$ is $\pi$. 3 MARKS
This shows the function repeats every $\pi$, so the period is $\pi$ [1].
Comprehensive answers (click to reveal)
Drill 1: Asymptotes at $x = -\frac{\pi}{2}, \frac{\pi}{2}$; intercept at $(0, 0)$. Increasing branches.
Drill 2: Asymptotes at $x = 0, \pi, 2\pi$; intercepts at $\frac{\pi}{2}, \frac{3\pi}{2}$. Decreasing branches.
Drill 3: Period = $\frac{\pi}{2}$. Asymptotes at $x = \frac{\pi}{4}, \frac{3\pi}{4}$. Intercepts at $0, \frac{\pi}{2}, \pi$.
Drill 4: Period = $\frac{\pi}{3}$. Asymptotes: $3x = n\pi \Rightarrow x = \frac{n\pi}{3}$.
Drill 5: $\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \tan x$. Since the function repeats every $\pi$, the period is $\pi$.
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Review and finish
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