
A unit circle is a circle with a radius of one (a unit radius). In trigonometry, the unit circle is centered at the origin.
For the point (x,y) in Quadrant I, the lengths x and y become the legs of a right triangle whose hypotenuse
is 1.
Using the right triangle and the Pythagorean Theorem, we can see that x^{2} + y^{2} = 1.
Thus, the equation of the unit circle is x^{2} + y^{2} = 1.




Note that x^{2} + y^{2} = 1 becomes cos^{2}θ + sin^{2}θ = 1. 
Four Quadrants and Unit Circle:
The placement of our right triangle can be in any of the four quadrants of the unit circle. While the measurements of the sides of the triangle are "positive" measurements, their locations take on the appropriate positive or negative signs of the quadrant.
For the point (x,y) in Quadrant 2, the side length of x is in a negative direction and the length y is in a positive direction. The hypotenuse remains positive 1 in all 4 quadrants. 


In Quadrant I, both x and y are positive.
In Quadrant II, x is negative and y is positive.
In Quadrant III, both x and y are negative.
In Quadrant IV, x is positive and y is negative.
In reference to the placement of our right triangle, remember that x represents the length of the horizontal leg and y represents the length of the vertical leg. 

Points of Special Interest on the Unit Circle
(in relation to the special right triangles of 30º60º90º and
45º45º90º)
(Download this picture .pdf)
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