We saw in the previous section that the unit circle is a circle with a radius of one, and sine and cosine are defined in terms of periodic movement around the unit circle. We measure distances around the unit circle in terms of phase. The phase can be thought of as the angle between our rotating line and the x-axis. We can express the phase in either degrees or radians.1 For any given point on the unit circle, the y-coordinate of the point is given by the sine of the phase, and the x-coordinate is given by cosine of the phase. You can fiddle with the slider in the bottom left corner of Figure 1 to better understand the relationship between the unit circle, sine, cosine, and phase.
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Figure 1. The Unit Circle
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If you’re willing to recall some high school trigonometry, you’ll probably remember that sine and cosine were defined using the mnemonic SOHCAHTOA,
Sine is Opposite over Hypotenuse Cosine is Adjacent over Hypotenuse Tangent is Opposite over Adjacent Recall further that opposite refers to the length of the vertical or green line, and adjacent refers to the length of the horizontal or red line.
sin(phase) = opposite / hypotenuse
cos(phase) = adjacent / hypotenuse In the case of the unit circle, the length of the hypotenuse (blue line) is always equal to one, making our equations pretty trivial.
sin(phase) = opposite / 1
cos(phase) = adjacent / 1 |
