Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns

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/

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!

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/