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Graphing Waves

Overview

The Rhythm of the Universe.

What is it?

Plotting trigonometric functions on a coordinate plane to create periodic waves.

History

Used by Fourier to prove that all sounds can be broken down into sums of simple sine waves.

Key Idea

Sine starts at 0, Cosine starts at 1 (the peak).

Practice This Topic

Concept Guide

Plain English: If you unzip the Unit Circle and lay it flat, you get a wave. This wave repeats forever. It represents anything that oscillates or cycles.

Real-world example: Audio Engineering. Sound waves, noise cancellation headphones, and music synthesizers all manipulate Sine waves.

How to do it

  1. Identify the Amplitude (number in front of the function).
  2. Identify the Period (2π divided by the number inside).
  3. Mark your starting point (0 for Sin, Max for Cos).
  4. Draw the wave curve up and down between the max and min amplitude.
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Common Pitfall

Confusing Amplitude (height of the wave) with Period (length of one cycle).

Word Problem
"The tide depth changes like a sine wave. High tide is 10ft, low tide is 2ft. What is the amplitude?"
Reasoning: The total difference is 8ft. The amplitude is half the distance from middle to peak. So, (10-2)/2 = 4ft amplitude.

Practice Examples

Sine Wave
1 | /--\ 0 |__/ \__ -1 | \--
Starts at 0, goes up to 1, down to -1.