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The Unit Circle

When you work with angles in all four quadrants, the trig ratios for those angles are computed in terms of the values of x, y, and r, where r is the radius of the circle that corresponds to the hypotenuse of the right triangle for your angle.

right triangle in second quadrant, with height y, base x, hypotenuse r labelled for angle (theta)

Any two right triangles with the same base angle θ ("theta")will be similar in the technical sense of having their sides in proportion. This similarity is more obvious when when the triangles are nested:

nested triangles; small with base 8.7, height 5.8; large with base 14.4, height 9.6

Similarity (and thus proportionality) means that the trig ratios from the two nested triangles displayed above will be the same:   Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

small triangle: sin(@) = 0.5547, cos(@) = 0.8321, etc; large triangle's values identical

The trig ratios for the same-sized angle θ are the same, even though the specific numbers from the two triangles' sets of sides are different. This emphasizes that, for trigonometric ratios, it's the angle θ that matters, not the particular triangle from which you obtained the angle.

To simplify computations, mathematicians like to fit an angle's triangle into a circle with radius r = 1. Once the hypotenuse has a fixed length of r = 1, then the trig ratios will depend only on x and y, since multiplying or dividing by r = 1 won't change anything. Only the values of x and y will matter.

circle with radius r = 1, overlaid on x,y-axes, with right triangle in first quadrant; base x, height y, hypotenuse r labelled

The Unit Circle

The point of the unit circle is that it makes other parts of mathematics easier and neater. You can even now understand some of how the unit circle can be useful. For instance, in the unit circle, you have, for any angle θ, the trig values for sine and cosine are clearly sin(θ) = y and cos(θ) = x. Working from this, you can take the fact that the tangent is defined as being tan(θ) = y/x, and then substitute for x and y to easily prove that the value of tan(θ) also must be equal to the ratio sin(θ)/cos(θ).

Another thing you can see from this is that the values of sine and cosine will never be more than 1 or less than –1, since x and y never take on values outside of this interval. Also, since tangent involves dividing by x, and since x = 0 when you're one-fourth and three-fourths of the way around the circle (that is, when you're at 90° and at 270°), the tangent will not be defined for these angle measures.

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Cite this article as:

Stapel, Elizabeth. "The Unit Circle." Purplemath. Available from Accessed


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