- Unit circle
mathematics, a unit circle is a circlewith a unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate systemin the Euclidean plane. The unit circle is often denoted "S"1; the generalization to higher dimensions is the unit sphere.
If ("x", "y") is a point on the unit circle in the first quadrant, then "x" and "y" are the lengths of the legs of a
right trianglewhose hypotenuse has length 1. Thus, by the Pythagorean theorem, "x" and "y" satisfy the equation
Since "x"2 = (−"x")2 for all "x", and since the reflection of any point on the unit circle about the "x"- or "y"-axis is also on the unit circle, the above equation holds for all points ("x", "y") on the unit circle, not just those in the first quadrant.
One may also use other notions of "distance" to define other "unit circles", such as the
Riemannian circle; see the article on mathematical norms for additional examples.
Trigonometric functions on the unit circle
trigonometric functions cosine and sine may be defined on the unit circle as follows. If ("x", "y") is a point of the unit circle, and if the ray from the origin (0, 0) to ("x", "y") makes an angle"t" from the positive "x"-axis, (where counterclockwise turning is positive), then
The equation "x"2 + "y"2 = 1 gives the relation
:Note that cos2(t)=(cos(t))2. This is the standard shorthand for expressing powers of trigonometric functions.
The unit circle also gives an intuitive way of realizing that
sineand cosineare periodic functions, with the identities
: : for any
These identities come from the fact that the "x"- and "y"-coordinates of a point on the unit circle remain the same after the angle "t" is increased or decreased by any number of revolutions (1 revolution = 2π radians = 360º).
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, using the unit circle, these functions have sensible, intuitive meanings for any real-valued angle measure.
In fact, not only sine and cosine, but all of the six standard trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like
versineand exsecant— can be defined geometrically in terms of a unit circle, as shown at right.
Complex numbers can be identified with points in the Euclidean plane, namely the number "a" + "bi" is identified with the point ("a", "b"). Under this identification, the unit circle is a group under multiplication, called the circle group. This group has important applications in mathematics and science.
* [http://www.dudefree.com/unitcircle/ An excellent Flash animation for learning the unit circle]
* [http://www.nbritton.org/uploads/unit_circle.pdf Printable, full page, unit circle handout]
* [http://glab.trixon.se/ GonioLab] : Visualization of the unit circle, trigonometric and hyperbolic functions
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Look at other dictionaries:
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unit circle — noun Date: 1955 a circle having a radius of 1 … New Collegiate Dictionary
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