Unit Circle : circle centered at the origin, and a radius of 1
x2 + y2 = 1
Since the radius in the unit circle is one and s =


To complete the unit circle it is important to remember our special right triangles: 30-60-90, 45-45-90
Their sides are related like this:
Once you remember that, it is easy to go on, simply drop an altitude from an point in the first quadrant, I will begin with the point
- 

Dropping an altitude from
creates a 30 - 60 - 90 triangle (
= 30). Since the hypotenuse equals 1 (because it's the radius), we saw above that the base(x) will equal half of that, so (1/2) and the height(y) will equal
or
.




It would look like this:
Again the hypotenuse it 1 because the radius is one meaning the both of the sides would equal
but in mathematics we rationalize that into
, both x and y would equal this.


This next point would look like this:
The last point is easiest because technically we've already done it. When you drop and altitude from
, it also creates a 30 - 60 - 90 triangle with a hypotenuse of one; however, this time the legs (x and y) have been switched.

The base (x) is now half the hypotenuse times the square root of three:
. And the height is half the radius so
.


It looks like this:

This is the Unit Circle, the rest is completed by manipulating this same number depending on which axis the first quadrant is reflected. The end product look like this:
Trig Functions:
Dividing two of the three sides of a right triangle will warrant a special result, no matter which sides are divided.
The functions Cosine (adjacent over hypotenuse) and Secant (hypotenuse over adjacent), are both even functions. This means that the function of -
, will yield a result
(the function being cosine or secant).


When graphed, these function are also reflections over the y - axis.
The functions Sine(opposite over hypotenuse), Cosecant (hypotenuse over opposite), Tangent (opposite over adjacent), Cotangent (djacent over opposite) are all odd functions. This means that the function of -
will yield a result of the -f(
).


When graphed, functions are also reflected over the x - axis.
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