The Unit Circle

The unit circle is simply a circle that is centered at the origin of a standard
coordinate plane, and has a radius of one

For the unit circle, we also know that half of a rotation is π , and that a full rotation is 2π 

And if we fill in the rest of the unit circle, it looks like this: 

But that's a lot to take in, so let's just focus on the first quadrant.
  

Okay, so where did we get all those points from? Let's think back to something we learned a few years ago. 

At some point in geometry, we learned about these things called.....

SPECIAL TRIANGLES! WOOOOOHOOOOOO

 

But, what does that have to do with the unit circle? Well, if we look back at Quadrant I, the different 
points are at 30°, 45°, and 60°. Coincidence? Probably not.




So, let's take a look at the 30/60/90 triangle first. 

In geometry we learned that we could put in the variable 2s for the hypotenuse, s for the side opposite of 30°, and s√3 for the side opposite of 60°.  

Now let's pretend that this triangle is in the unit circle. That means that the hypotenuse would have to be 1, right? So we set 2s equal to 1 and solve for s. 

 

    2s = 1

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2s/2 = 1/2

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s = 1/2

 

Once we get s, we can plug that in and solve for the other sides, so we end up with:

And if we put this into the unit circle, we would see this: 

The sides of the triangle are the x and y coordinates on the circle at 60°.

The unit circle at 30° makes the same triangle as the one above, it's just flipped and rotated. 

Alright, now the second special triangle is the 45/45 triangle. Geometry taught us that we could put in the variable s√2 for the hypotenuse. And because the 45/45 is an isosceles triangle, we put in s for both sides. 

Again, we want the hypotenuse to equal 1, so we make s√2 = 1 and solve. 

s√2 = 1

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s√2/√2 = 1/√2

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s = (1/√2)  * (√2/√2)

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s = √2/√4 

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s = √2/2

Again, we plug in √2/2 for s, and we get this:

And putting this on the unit circle, we see that the same goes for the 45/45 triangle.

So now we understand the math behind the unit circle YAYYYYYY!