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Section 4.8
Graphs of Other
Trig Functions
Objective: Sketch the graphs of tangent, cotangent, secant, and cosecant functions.
Tangent & Cotangent
Understanding the parent graphs
Concept

Graph \(y = \tan x\)

\(x\)\(\tan x\)
\(0\)\(0\)
\(\pi/4\)\(1\)
\(\pi/2\)und
\(3\pi/4\)\(-1\)
\(\pi\)\(0\)
\(5\pi/4\)\(1\)
\(3\pi/2\)und
\(7\pi/4\)\(-1\)
\(2\pi\)\(0\)
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Concept

Properties of Tangent

Period
The period is the distance between two consecutive asymptotes.
Amplitude
The amplitude is undefined, but you can have a vertical stretch.
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Concept

Graph \(y = \cot x\)

\(x\)\(\cot x\)
\(0\)und
\(\pi/4\)\(1\)
\(\pi/2\)\(0\)
\(3\pi/4\)\(-1\)
\(\pi\)und
\(5\pi/4\)\(1\)
\(3\pi/2\)\(0\)
\(7\pi/4\)\(-1\)
\(2\pi\)und
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Concept

Properties of Cotangent

Period
The period is the distance between two consecutive asymptotes.
Amplitude
The amplitude is undefined, but you can have a vertical stretch.
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Reference

Basic Characteristics of Tangent & Cotangent

\(f(x) = \tan(x)\) \(f(x) = \cot(x)\)
Domain \(x \neq \frac{\pi}{2} + n\pi\) \(x \neq n\pi\)
Range All real numbers
Period \(\pi\)
\(x\)-intercepts \((n\pi, 0)\) \(\left(\frac{\pi}{2} + n\pi,\, 0\right)\)
\(y\)-intercept \((0, 0)\) None
Vertical Asymptotes \(x = \frac{\pi}{2} + n\pi\) \(x = n\pi\)
Even / Odd Odd
Symmetry About the origin
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Key Idea

The 5 Key Points from Parent Graphs

\(y = \tan x\)
\(x\)\(\tan x\)
\(-\pi/2\)undasymptote
\(-\pi/4\)\(-1\)low
\(0\)\(0\)inflection
\(\pi/4\)\(1\)high
\(\pi/2\)undasymptote
\(y = \cot x\)
\(x\)\(\cot x\)
\(0\)undasymptote
\(\pi/4\)\(1\)high
\(\pi/2\)\(0\)inflection
\(3\pi/4\)\(-1\)low
\(\pi\)undasymptote
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Method

Steps for Graphing a Tangent or Cotangent Function

Step 1 — T-Table
Write down a t-table with 5 points for the parent graph.
Step 2 — Analyze
Determine the transformations of the function.
affects \(x\)-coordinates
period
phase shift
→
affects \(y\)-coordinates
vertical stretch
vertical translation
Steps 3 & 4 — New Coordinates
New \(x\): Use the period, then phase shift to adjust the \(x\)-coordinates.
New \(y\): Use the vertical stretch, then vertical translation to adjust the \(y\)-coordinates.
Step 5 — Plot & Draw
Set up axes, scale, plot points, and draw the graph!
The order matters: multiply before add/subtract!
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Example 1

Sketch two periods of \(y = \tan(2x) + 3\)

Parent t-table & Analysis:
\(x\)\(\tan x\) new \(x\) new \(y\)
\(-\pi/2\)und \(-\pi/4\) und
\(-\pi/4\)\(-1\) \(-\pi/8\) \(2\)
\(0\)\(0\) \(0\) \(3\)
\(\pi/4\)\(1\) \(\pi/8\) \(4\)
\(\pi/2\)und \(\pi/4\) und
Period: \(\pi/2\) → multiply \(x\)'s by \(1/2\)
Phase shift: none
Vertical stretch: 1 (none)
Vertical translation: \(+3\)
\(x\): multiply by \(1/2\)
\(y\): add \(3\)
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Example 2

Sketch the graph of \(y = 2\cot(x/3) - 1\)

Parent t-table & Analysis:
\(x\)\(\cot x\) new \(x\) new \(y\)
\(0\)und \(0\) und
\(\pi/4\)\(1\) \(3\pi/4\) \(1\)
\(\pi/2\)\(0\) \(3\pi/2\) \(-1\)
\(3\pi/4\)\(-1\) \(9\pi/4\) \(-3\)
\(\pi\)und \(3\pi\) und
Period: \(3\pi\) → multiply \(x\)'s by \(3\)
Phase shift: none
Vertical stretch: \(2\) → multiply \(y\)'s by \(2\)
Vertical translation: \(-1\) → subtract \(1\) from \(y\)'s
\(x\): multiply by \(3\)
\(y\): multiply by \(2\), then subtract \(1\)
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Cosecant
The reciprocal of sine
Concept

Graphing Cosecant from Sine

To graph the cosecant you need to take every point on the sine curve and take the reciprocal.
\(y = \sin(x)\) shown dashed
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Secant
The reciprocal of cosine
Concept

Graphing Secant from Cosine

To graph the secant you need to take every point on the cosine curve and take the reciprocal.
\(y = \cos(x)\) shown dashed
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Reference

Basic Characteristics of Secant & Cosecant

\(f(x) = \csc(x)\) \(f(x) = \sec(x)\)
Domain \(x \neq n\pi\) \(x \neq \frac{\pi}{2} + n\pi\)
Range \((-\infty, -1] \cup [1, \infty)\)
Period \(2\pi\)
Intercepts None \((0, 1)\)
Vertical Asymptotes \(x = n\pi\) \(x = \frac{\pi}{2} + n\pi\)
Even / Odd Odd Even
Symmetry About the origin About the \(y\)-axis
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Method

Steps for Graphing a Secant or Cosecant Function

Step 1 — Analyze the Reciprocal
Analyze the function's reciprocal function. Find the vertical/horizontal dilation, reflection, or translation.
Step 2 — T-Table
Make the t-table for the reciprocal function.
Step 3 — Identify Asymptotes
Identify the vertical asymptotes. This is where the reciprocal function (sine or cosine) would cross its midline.
Step 4 — Plot & Sketch
Plot the asymptotes and other points from the t-table and sketch the graph.
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Example 3

Sketch the graph of \(y = 2\csc\!\left(x + \frac{\pi}{4}\right)\)

Reciprocal: \(y = 2\sin\!\left(x + \frac{\pi}{4}\right)\)
Vertical stretch: \(2\) → multiply \(y\)'s by \(2\)
Period: \(2\pi\) (no change)
Phase shift: \(-\pi/4\) (left \(\pi/4\))
\(x\)\(\sin x\) new \(x\) new \(y\)
\(0\)\(0\) \(-\pi/4\) \(0\) (asym)
\(\pi/2\)\(1\) \(\pi/4\) \(2\)
\(\pi\)\(0\) \(3\pi/4\) \(0\) (asym)
\(3\pi/2\)\(-1\) \(5\pi/4\) \(-2\)
\(2\pi\)\(0\) \(7\pi/4\) \(0\) (asym)
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Example 4

Sketch the graph of \(y = -2\sec(\pi x - \pi) + 1\)

Reciprocal: \(y = -2\cos(\pi x - \pi) + 1\)
Vertical stretch: \(-2\) → multiply \(y\)'s by \(-2\)
Vertical translation: \(+1\) → add \(1\) to \(y\)'s
Period: \(2\pi/\pi = 2\)
Phase shift: \(\pi/\pi = 1\) (right \(1\))
\(x\)\(\cos x\) new \(x\) new \(y\)
\(0\)\(1\) \(1\) \(-1\)
\(\pi/2\)\(0\) \(1.5\) \(1\) (asym)
\(\pi\)\(-1\) \(2\) \(3\)
\(3\pi/2\)\(0\) \(2.5\) \(1\) (asym)
\(2\pi\)\(1\) \(3\) \(-1\)
\(y\): multiply by \(-2\), then add \(1\)
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Homework
Graphs of Other Trig Functions CIRCUIT