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. | ||
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>. | 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{opposite}{hypotenuse}</math> and <math>\cos x = \frac{near}{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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Revision as of 18:41, 12 October 2007
| This is an AoPSWiki Word of the Week for Oct 11-17 |
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.
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