Angles, Theta, and Sine Waves

The idea, in my own words.

So last time we talked about the function of trigonometric functions but what about the trigonometric part? It's taking an angle in a right triangle and calling it theta. Now that we have theta, we relate it to the ratio of two sides of the triangle. The ratio opposite over hypotenuse, which looks like vertical over diagonal in the triangle below, is what we call sine. When you change θ and track sine across all those angles, you get a wave shape.

Worked example.

The function used for these waves is f(θ) = a · sin(θ), where a tells us how tall the wave gets. If a = 1, the wave goes from 1 to negative 1. If a = 2, it goes from 2 to negative 2. Below, I'm plugging in a few values for θ to see the wave shape come out of the math.

• θ = 0°, sin(θ) = 0
• θ = 30°, sin(θ) = 0.5
• θ = 60°, sin(θ) = 0.87
• θ = 90°, sin(θ) = 1

This is the pattern. Pick a θ, get a height. Pick the next θ, get the next height. String them together and you have a sine wave.

Carried question.