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Encyclopedia > Unit circle
Illustration of a unit circle. t is an angle measure.

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 For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ...

$x^2 + y^2 = 1 ,!$

Since x2 = (−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. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

## Trigonometric functions on the unit circle GA_googleFillSlot("encyclopedia_square");

Relationship of 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 Image File history File links Unit_circle_angles. ... Image File history File links Unit_circle_angles. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about angles in geometry. ...

$cos(t) = x ,!$
$sin(t) = y ,!$

The equation x2 + y2 = 1 gives the relation

$cos^2(t) + sin^2(t) = 1 ,!$

The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...

$cos t = cos(2pi k+t) ,!$
$sin t = sin(2pi k+t) ,!$
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). The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

All of the trigonometric functions can be constructed geometrically in terms of a unit circle centered at O.

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. Define several trig functions from unit circle. ... Define several trig functions from unit circle. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

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 versine and exsecant — can be defined geometrically in terms of a unit circle, as shown at right. The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ...

## 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 under multiplication, called the circle group. This group has important applications in math and science; see circle group for more details. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (âˆ’1). ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ...

Results from FactBites:

 Unit circle - definition of Unit circle in Encyclopedia (402 words) In mathematics, a unit circle is a circle with 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 system in the Euclidean plane. Under this identification, the unit circle is a group under multiplication, called the circle group.
 Circle - Free Encyclopedia (834 words) All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. A part of a circle bound by two radii is called an arc, and the ratio between the length of an arc and the radius defines the angle between the two radii in radians. The 3-dimensional analog of the circle is the sphere.
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