Difference between revisions of "Unit circle"

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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>.  
 
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>.  
  
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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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>.
  
 
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 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 22:07, 12 October 2007

This is an AoPSWiki Word of the Week for Oct 11-17
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.

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