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 ==
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[[Image:Unit circle with triangle.png|left|400px|thumb]]
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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 ==
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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>.
  
 
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[[Category:Geometry]]

Revision as of 11:17, 18 October 2007

Unit circle.png

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

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

Trigonometry

Unit circle with triangle.png

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 $\theta$. If we drop a perpendicular from the point of intersection between the ray and the circle, we have a right triangle with hypotenuse of $1$.

Using the definitions $\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos x = \frac{\text{near}}{\text{hypotenuse}}$, we find that $\sin \theta = \frac{y}{1} = y$ and $\cos \theta = \frac{x}{1} = x$.

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: $\sin^2{\theta}+\cos^2{\theta}=1$. The proof is as follows:


We see that the length and width of the triangle in the diagram at left are $\sin{\theta}$ and $\cos{\theta}$, respectively. We use the pythagorean theorem to get:

$\sin^2{\theta}+\cos^2{\theta}=1$

Complex numbers

On the complex plane, all solutions to the polynomial $x^n = 1$ 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 $\cos \left(\frac{2\pi k}{n}\right) + i\sin \left(\frac{2\pi k}{n}\right)$, where $k = 0,1,2,\ldots n-1$. This is commonly written as $\mathrm{cis} \left(\frac{2\pi k}{n}\right)$.

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