# Difference between revisions of "Unit circle"

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− | + | [[Image:Unit_circle.png|right|300px|thumb]] | |

A '''unit circle''' is a [[circle]] whose [[radius]] has length 1. | A '''unit circle''' is a [[circle]] whose [[radius]] has length 1. | ||

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+ | In the Cartesian coordinate system, an equation of the form <math>(x-h)^2+(y-k)^2=1</math> defines a unit circle with center <math>(h,k)</math>. | ||

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+ | == Trigonometry == | ||

+ | [[Image:Unit circle with triangle.png|left|400px|thumb]] | ||

+ | An unit circle centered at the origin can be used to calculate values for the basic trigonometric functions. Suppose we draw a ray starting from the origin and meeting the positive x-axis with an angle of <math>\theta</math>. If we drop a perpendicular from the point of intersection between the ray and the circle, we have a right triangle with [[hypotenuse]] of <math>1</math>. | ||

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+ | Using the definitions <math>\sin x = \frac{\text{opposite}}{\text{hypotenuse}}</math> and <math>\cos x = \frac{\text{near}}{\text{hypotenuse}}</math>, we find that <math>\sin \theta = \frac{y}{1} = y</math> and <math>\cos \theta = \frac{x}{1} = x</math>. | ||

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+ | We can read off values for sine and cosine of an angle this way; we can draw the angle and approximate the x and y coordinates of the intersection. | ||

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+ | We can also prove one of he fundamental theorems of [[trigonometry]]: <math>\sin^2{\theta}+\cos^2{\theta}=1</math>. The proof is as follows: | ||

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+ | We see that the length and width of the triangle in the diagram at left are <math>\sin{\theta}</math> and <math>\cos{\theta}</math>, respectively. We use the [[pythagorean theorem]] to get: | ||

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+ | <math>\sin^2{\theta}+\cos^2{\theta}=1</math> | ||

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+ | == Complex numbers == | ||

+ | On the [[complex plane]], all solutions to the polynomial <math>x^n = 1</math> lie upon the unit circle. These are referred to as the [[roots of unity]]. | ||

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+ | In polar form, the solution to this polynomial can be expressed as <math>\cos \left(\frac{2\pi k}{n}\right) + i\sin \left(\frac{2\pi k}{n}\right)</math>, where <math>k = 0,1,2,\ldots n-1</math>. This is commonly written as <math>\mathrm{cis} \left(\frac{2\pi k}{n}\right)</math>. | ||

{{stub}} | {{stub}} | ||

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+ | [[Category:Geometry]] |

## Revision as of 11:17, 18 October 2007

A **unit circle** is a circle whose radius has length 1.

In the Cartesian coordinate system, an equation of the form defines a unit circle with center .

## Trigonometry

An unit circle centered at the origin can be used to calculate values for the basic trigonometric functions. Suppose we draw a ray starting from the origin and meeting the positive x-axis with an angle of . If we drop a perpendicular from the point of intersection between the ray and the circle, we have a right triangle with hypotenuse of .

Using the definitions and , we find that and .

We can read off values for sine and cosine of an angle this way; we can draw the angle and approximate the x and y coordinates of the intersection.

We can also prove one of he fundamental theorems of trigonometry: . The proof is as follows:

We see that the length and width of the triangle in the diagram at left are and , respectively. We use the pythagorean theorem to get:

## Complex numbers

On the complex plane, all solutions to the polynomial lie upon the unit circle. These are referred to as the roots of unity.

In polar form, the solution to this polynomial can be expressed as , where . This is commonly written as .

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