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>. | + | 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. | ||
{{stub}} | {{stub}} | ||
Revision as of 23:07, 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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