# Sine and Cosine Waves

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You will first notice a circle in the diagram at right. Now, notice that the green line equals the radius of that circle which is 1. Now, notice that the red line forms a right angle with the *x* axis. You can also see that the green line forms the hypotenuse of a right triangle.

Recalling the word SOHCAHTOA, you will notice that the sine of the angle at the origin (0, 0) equals ^{opposite}/_{hypotenuse} or ^{y}/_{1}. You can also deduce that the cosine of the angle at the origin (0, 0) equals ^{adjacent}/_{hypotenuse} or ^{x}/_{1}. Therefore the sine equals *y*, and the cosine equals *x* if the radius equals 1.

You remember that the circumference of a circle equals 2π*r*. So half the circumference equals π times the radius. Notice that the *x* values in the diagram are denoted as π and negative π. The distance between π and negative π is 2π, because the circumference of a circle with *r* = 1 is 2π.

## What Does a Circle Have To Do with a Sine Wave?

When we graph a sine wave, we are plotting the *x* and *y* values starting at zero and moving to the right instead of sweeping constantly around the origin as we do in a circle. By doing this we will have exactly one *y* that corresponds to a given *x* which is the definition of a function. The result is a graph like the one below.

Likewise, when graphing a cosine wave, we get a graph that looks like the one below. Notice that the cosine wave starts at *y* = 1 because the amplitude of the cosine wave corresponds to the length of the horizontal leg of the right triangle. The sine wave starts at *y* = 0 because the amplitude of the sine wave corresponds to the vertical leg of the right triangle.

**Just remember that the sine wave starts and y=0 and that the cosine wave starts at y=1.**

The animated graphic below demonstrates how a geometric figure (circle) becomes a function (wave) by changing the method of plotting.

The animated graphic below demonstrates the same process with a cosine wave.