What is the unit circle and why is it useful?
In: Mathematics
A unit circle is a circle whose radius is exactly 1 unit. They’re not really more useful than any other circle, but they are more convenient and easier to work with than any other circle. Because the radius is 1, you can just drop the r in most of the circle’s formulas (area, perimeter, etc). When using trig, functions, you can use the results as-is without having to multiply by r (effectively, the trig. functions are based on right triangles whose hypotenuse is exactly 1 unit. When applying them to arcs, the radius of the circle acts as the hypotenuse. Unit circle = unit hypotenuse = easy trig. functions).
The unit circle is way more useful, at least from an intuitive perspective. Because the hypotenuse of the right triangle inscribed by the edge of that circle and the origin is always length one, it gives you an excellent understanding of what trig functions actually mean.
Is that car heading off at a 30 degree angle going 100 miles per hour? You can immediately know that it’s going 50 miles per hour along the Y axis and ~87 miles per hour along the X axis, because the sine and cosine of 30 degrees is 0.5 and 0.866, respectively. Tank armor sloped at 45 degrees can be considered to be (1/0.707) times thicker than perfectly vertical armor to impacting shells because sine of 45 degrees is 0.707. Every particle of rocket exhaust not traveling directly away down the axis of thrust is wasting cosine(angle away from axis) percent of its momentum.
The thing about the unit circle is that it’s a template to give you the proportions, and then you fill in the magnitude later to get an answer.
The formula for a circle is x^2 + y^2 = r^2 . With a little bit of effort, we can transform this into trigonometric functions: x^2 / r^2 + y^2 / r^2 = 1, then cosine^2 + sine^2 = 1. In a unit circle, the radius is 1 unit long, so we can drop it from the formulas, and cos and sin of the various angles just become their x and y co-ordinates.
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