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<!--Hey can we get a picture (including the tangent values) of the points of interest on the unit circle with the tangent values shown also?-->[[Image:Unit circle.svg|Unit circle|right|thumb|186px|Illustration of a unit circle. The variable ''t'' is an [[angle]] measure.]]
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In [[mathematics]], a '''unit circle''' is a [[circle]] with a [[radius]] of [[1 (number)|one]].  Frequently, especially in [[trigonometry]], the unit circle is the circle of radius one centered at the origin (0, 0) in the [[Cartesian coordinate system]] in the [[Euclidean plane]]. The unit circle is often denoted ''S''<sup>1</sup>; the generalization to higher dimensions is the [[unit sphere]].
 
If (''x'', ''y'') is a point on the unit circle, 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
 
:<math>x^2 + y^2 = 1.</math>
 
Since ''x''² = (&minus;''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 only 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 [[norm (mathematics)|mathematical norms]] for additional examples.
 
==In the complex plane==
The unit circle can be considered as the set of [[complex numbers]] ''z'' of the form
:<math> z = \,\mathrm{e}^{i t}\, = \cos(t) + i \sin(t)  \,</math>
for all ''t''. This relation is [[Euler's formula]].
 
==Trigonometric functions on the unit circle==
[[Image:Circle-trig6.svg|right|thumb|300px|''All'' of the trigonometric functions of the angle ''θ'' can be constructed geometrically in terms of a unit circle centered at ''O''.]]
[[File:Periodic sine.PNG|thumb|Sine function on unit circle (top) and its graph (bottom)]]
The [[trigonometric function]]s 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
 
:<math>\cos(t) = x \,\!</math>
:<math>\sin(t) = y. \,\!</math>
 
The equation ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1 gives the relation
 
:<math> \cos^2(t) + \sin^2(t) = 1. \,\!</math>
 
The unit circle also demonstrates that [[sine]] and [[cosine]] are [[periodic function]]s, with the identities
 
:<math>\cos t = \cos(2\pi k+t) \,\!</math>
:<math>\sin t = \sin(2\pi k+t) \,\!</math>
for any [[integer]] ''k''.
 
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(''x''<sub>1</sub>,''y''<sub>1</sub>) on the unit circle such that an angle ''t'' with 0 < ''t'' < π/2 is formed with the positive arm of the ''x''-axis. Now consider a point Q(''x''<sub>1</sub>,0) and line segments PQ <math>\perp</math> OQ. The result is a right triangle ΔOPQ with ∠QOP = ''t''. Because PQ has length ''y''<sub>1</sub>, OQ length ''x''<sub>1</sub>, and OA length 1, sin(''t'') = ''y''<sub>1</sub> and cos(''t'') = ''x''<sub>1</sub>. Having established these equivalences, take another radius OR from the origin to a point  R(−''x''<sub>1</sub>,''y''<sub>1</sub>) on the circle such that the same angle ''t'' is formed with the negative arm of the ''x''-axis. Now consider a point S(''−x<sub>1</sub>'',0) and line segments RS <math>\perp</math> OS. The result is a right triangle ΔORS with ∠SOR = ''t''. It can hence be seen that, because ∠ROQ = π−''t'', R is at (cos(π−''t''),sin(π−''t'')) in the same way that P is at (cos(''t''),sin(''t'')). The conclusion is that, since  (−''x''<sub>1</sub>,''y''<sub>1</sub>) is the same as (cos(π−''t''),sin(π−''t'')) and (''x''<sub>1</sub>,''y''<sub>1</sub>) is the same as (cos(''t''),sin(''t'')), it is true that sin(''t'') = sin(π−''t'') and −cos(''t'') = cos(π−''t''). It may be inferred in a similar manner that tan(π−''t'') = −tan(''t''), since tan(''t'') = ''y''<sub>1</sub>/''x''<sub>1</sub> and tan(π−''t'') = ''y''<sub>1</sub>/(−''x''<sub>1</sub>). A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/sqrt(2).
 
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, when defined with the unit circle, these functions produce meaningful values for any [[real number|real]]-valued angle measure&nbsp;– even those greater than 2π. In fact, all six standard trigonometric functions&nbsp;– sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like [[versine]] and [[exsecant]]&nbsp;– can be defined geometrically in terms of a unit circle, as shown at right.
 
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the [[Trigonometric identity#Angle_sum_and_difference_identities|Sum and Difference Formulas]].
 
[[Image:Unit circle angles color.svg|thumb|300px|left|The unit circle, showing [[Exact trigonometric constants|coordinates of certain points]]]] <!--Get a picture with the tangent values show-->
 
==Circle group==
[[Complex number]]s 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)|group]] under multiplication, called the [[circle group]]. On the plane multiplication by <math>\cos \theta + i \sin \theta</math> gives a counterclockwise rotation by &theta;. This group has important applications in mathematics and science.{{Such as?}}
 
==Complex dynamics==
{{Main|Complex dynamics}}
[[Image:Erays.png|right|thumb|Unit circle in complex dynamics]]
[[Julia set]] of [[Dynamical system (definition)|discrete nonlinear dynamical system]] with [[evolution function]]:
 
:<math>f_0(x) = x^2 \,</math>
 
is a unit circle. It is a simplest case so it is widely used in study of dynamical systems.
 
==See also==
*[[Angle|Angle measure]]
*[[Circle group]]
*[[Pythagorean trigonometric identity]]
*[[Riemannian circle]]
*[[Unit disc]]
*[[Unit hyperbola]]
*[[Unit square]]
*[[Z-transform]]
 
==External links==
{{Wikibooks|Trigonometry/The unit circle}}
{{Wiktionary|unit circle}}
*{{mathworld | urlname = UnitCircle | title = Unit circle}}
*[http://www.dudefree.com/unitcircle/ Flash animation for learning the unit circle]
*[http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions
 
[[Category:Circles]]
[[Category:One]]
[[Category:Trigonometry]]
[[Category:Fourier analysis]]
[[Category:Analytic geometry]]

Latest revision as of 06:45, 22 July 2014

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I feel that selecting a identify brand knife is the ideal way to go. When acquiring a knife , you want to be sure that the blade is made from a great quality of steel, so it will hold an edge, and sharpen well when necessary. Most pocket knives are utilized on a daily bases and want to be in a position to keep up to vigorous everyday use and tear with out failure. Plain edge or serrated? A simple edge knife is simpler to sharpen and doesn't demand any specific sharpening stones. Even so the serrated edge is excellent at cutting tinder or rope. Partially serrated edges operate greater on longer blades.
All round I actually like the Folding Utility Ceramic Knife from Flint Knives and hugely suggest this knife or the smaller pocket knife as an all all around multipurpose knife for property and area use. The blade is sharp and will continue to be that way with appropriate care. It is not tough to sharpen as prolonged as you use the correct instruments and the knife is great as nicely, I suggest to have a ceramic knife in the first area is neat. Several men and women since their childhood once or two times it happens to occur across to pocket knives use. In some families gifting knives on situations is genuinely a typical apply. read far more
The new Rapidly Draw Tanto is a folder with Gerber’s Assisted Opening 2. system, which provides sleek opening action right after a brief split-in interval (it’s a minor rigid correct out of the deal). The 3-inch blade is fashioned from 7Cr17MoV metal and has twin thumb studs for proper or remaining-handed opening. The tanto stage is best for penetration, and the serrated segment of the blade allows for rapid, aggressive cutting. The abnormal locking mechanism is a plunge lock, made to keep your fingers out of the path of the blade when closing. The nylon handle, with textured scales, provides a secure grip with out including fat.
A lot of hunters set a good deal of believed into the blade design and style of their searching knife, but put really tiny believed into the substance of the deal with. The traditional wooden, bone, or leather handles are really practical and interesting to the eye. Even so, don't overlook the more recent handle resources, though not as pleasing to the eye, rubber and other composites merit a seem. The newer handle resources offer you tremendously enhanced manage in adverse situations giving the hunter a higher diploma of safety. If you are looking for for prolonged survival, then you are recommended to get a multi-goal blade that evidently suggests that the blade is a portion straight edge.
There are two principal sorts of hunting knives to decide on from, they are the fixed blade knife and folding or lock blade knife Each designs have their possess unique benefits. When picking between the two you will have to decide which type will suit your specific searching style. This determination will be dependent on your individual preference and how you desire to have your knife , on your belt or in a pack. Usually remember that a folding knife IS a weapon. Utilized with drive, you could stop up bodily hurting somebody, even by yourself. Additionally, you may possibly even lead to unneeded pain and harm to one more person or his home.
Remaining side,” the boy said. “I got a bang on the head a even though back and I can’t hear a lot with my left ear.” Renno grinned. “Your mum nevertheless into the witchcraft stuff?” “It’s a soreness,” the boy mentioned regrettably. “Of training course, she in no way makes use of any of it herself, just tries it out on other people. One particular of the maids almost died. You can see why I maintain tranquil about my ear.” “Better deaf than dead,” Renno agreed solemnly. “What took place?” When Renno shrugged, there was a recommendation of making an attempt to shake off an frustrating entanglement. “And absolutely nothing,” he mentioned. “I informed you all I know.”

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