Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

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The unit circle is a circle that has a radius of one and is centered at the origin of the coordinate plane. It is a concept that frequently occurs in many of the math subjects, especially those where Trigonometry is used. Questions asking about **unit circle coordinates** often give an unknown coordinate and require us to use the properties of a unit circle to calculate these coordinates.

The unit circle is something that we're going to start talking about in Geometry, we're going to talk about it in Algebra too, and you're going to talk about it in pre-calc and probably a little bit of Calculus as well. So you may as well get familiar with it right now, let's start off with what is the unit circle? Well unit in Math usually implies the number 1, so the unit circle is the circle with the radius of 1 centered at the origin. So right here I've drawn a circle that's centered at the origin and if the radius is 1 we can draw in a couple of key points. I know that this point right here where my circle intercepts the x axis is going to be 1,0 I know at this point right here, where it intercects the y axis is going to be at 0 and 1. So usually on your homework or a quiz when you have the unit circle they're going to let you know that it is a unit circle by indicating these 2 points.

We also know at this point right here just for the sake of information, is going to be at negative 1, 0 and this point right here is going to be at 0, negative 1. So how does this relate to the Pythagorean Theorem? Well you're probably going to have a problem where they draw in a radius, so I guess we could say draw in a radius r, so what you're going to do, is you're going to say well r since the radius of this circle is 1 will be 1 and to find the value of r based on some point x and y what you're going to do is you're going to drop an altitude. So I'm going to grab a different color marker here and you're going to drop an altitude all the way down to your x axis, creating a right triangle. So if I were to re-draw this triangle down here, your hypotenuse which is going to be opposite your right angle is going to be r which is 1. 1 leg is going to be your x coordinate and the other leg is going to be your y coordinate.

Now when you get to the second, third and fourth quadrants here where x is negative, here where x and y are negative and the fourth where y is negative, you're going to use the absolute value of x and y because you want to have positive numbers. So the key to using the unit circle is to remember that your radius will be and that you can always drop an altitude or if you're down here raise an altitude so you create a right triangle.