Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns

by Justin Skycak (@justinskycak) on May 12, 2019

Equations involving compositions of trigonometric functions can create wild patterns in the plane.

This post is part of the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns. In Graphing Calculator Drawing Exercises. https://justinmath.com/graphing-calculator-drawing-composition-waves-implicit-trig-patterns/

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Setup. Navigate to https://www.desmos.com/calculator. Be sure to sign in so that you can save your graph.

Demonstration - Composition Waves. Observe the graph as you type each of the following inputs.

$\begin{align*} y=\sin(x) \end{align*}$

$\begin{align*} y=\sin(x^2) \end{align*}$

$\begin{align*} y=\sin\left( \frac{1}{x} \right) \end{align*}$

$\begin{align*} y=\sin\left( \tan(x) \right) \end{align*}$

$\begin{align*} y=\tan\left( \sin(x) \right) \end{align*}$

Demonstration - Implicit Trig Patterns. Observe the graph as you type each of the following inputs.

$\begin{align*} \sin(y)=\cos(x) \end{align*}$

$\begin{align*} \sin(y)=\sin\left(x+y\right) \end{align*}$

$\begin{align*} \sin\left(y\right)\tan\left(x\right)=\sin\left(x+y\right) \end{align*}$

$\begin{align*} \sin\left(y\right)\tan\left(x\right)=\cos\left(x\right)\tan\left(y\right) \end{align*}$

$\begin{align*} \sin\left(y\right)\cos\left(x\right)=0.1 \end{align*}$

$\begin{align*} \sin\left(xy\right)=\sin\left(x+y\right) \end{align*}$

Challenge. Create some interesting wallpapers using implicit trig patterns!

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