#redirect Unit circle
[[Image:Unit_circle.png|Unit circle|right|frame|Illustration of a unit circle. t is an angle measure.
:Image:UnitCircle.png]] In mathematics, a unit circle is a circle with 1 (number) 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 system in the Euclidean plane. The unit circle is often denoted ''S''1. 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 triangle whose 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"; see the article on normed vector space for examples. ==Trigonometric functions on the unit circle== The 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 the angle is measured in the counter-clockwise direction), then : : The equation ''x''2 + ''y''2 = 1 gives the relation : The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities : : :for any integer ''k''. 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). 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 number-valued angle measure. ==Circle group== 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 (mathematics) under multiplication, called the circle group. This group has important applications in math and science; see circle group for more details. ==See also== *Trigonometric function *Angle *Unit square *Circle group Trigonometry
This was a duplicate page, at "Unit Circle": ----------------------------- The Unit Circle is a circle of radius 1 unit, oftenly used to define the functions of trigonometry. In this diagram, individual points on the unit circle are labeled first with its coordinates (exact values), with the angle in degree angular measure, then with radian angular measure. Points in the lower hemisphere have both positive and negative angles marked. ---- Geometers notation S2: really? I don't think I have ever seen this - which is not to say it isn't used, but is it common enough to warrant inclusion? User:Charles Matthews 12:30, 17 Aug 2004 (UTC)
See other meanings of words starting from letter:
Words begining with Unit_circle:
Unit_Circle
Unit_circle
Unit_circle
Unit Circle
#redirect Unit circle
Unit circle
[[Image:Unit_circle.png|Unit circle|right|frame|Illustration of a unit circle. t is an angle measure.
:Image:UnitCircle.png]] In mathematics, a unit circle is a circle with 1 (number) 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 system in the Euclidean plane. The unit circle is often denoted ''S''1. 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 triangle whose 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"; see the article on normed vector space for examples. ==Trigonometric functions on the unit circle== The 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 the angle is measured in the counter-clockwise direction), then : : The equation ''x''2 + ''y''2 = 1 gives the relation : The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities : : :for any integer ''k''. 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). 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 number-valued angle measure. ==Circle group== 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 (mathematics) under multiplication, called the circle group. This group has important applications in math and science; see circle group for more details. ==See also== *Trigonometric function *Angle *Unit square *Circle group Trigonometry
Unit circle
This was a duplicate page, at "Unit Circle": ----------------------------- The Unit Circle is a circle of radius 1 unit, oftenly used to define the functions of trigonometry. In this diagram, individual points on the unit circle are labeled first with its coordinates (exact values), with the angle in degree angular measure, then with radian angular measure. Points in the lower hemisphere have both positive and negative angles marked. ---- Geometers notation S2: really? I don't think I have ever seen this - which is not to say it isn't used, but is it common enough to warrant inclusion? User:Charles Matthews 12:30, 17 Aug 2004 (UTC)
See other meanings of words starting from letter:
U
UA | UB | UC | UD | UE | UF | UG | UH | UI | UJ | UK | UL | UM | UN | UO | UP | UR | US | UT | UW | UX | UY | UZ |Words begining with Unit_circle:
Unit_Circle
Unit_circle
Unit_circle
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