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 x2 + y2 = 1.  
Thus, the equation of the unit circle is x2 + y2 = 1.  

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If we examine angle θ (as shown at the left)
in this unit circle, we see that
3
which show us that in a unit circle,
4
also creating the ordered pair
5
Sine is represented by the vertical leg.
Cosine is represented by the horizontal leg.

Note  that     x2 + y2 = 1   becomes    cos2θ + sin2θ = 1.

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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.

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unitcircle4

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.


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Points of Special Interest on the Unit Circle
(in relation to the special right triangles of 30º-60º-90º and 45º-45º-90º)
(Download this picture .pdf)
unitcircleangles


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