The Unit Circle

The Unit Circle 

 

What is the Unit Circle?

The unit circle is a circle whose center is at the origin with a radius of one. Because the radius is 1, you can directly measure sine and cosine. If a point on the circle is on the terminal side of an angle in standard position, then the sine of such an angle is the y-coordinate of the point, and the cosine of the angle is the x-coordinate of the point.
 

 

The Pythagorean Theorem states that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:

 

x²+ y² = 1²

But 1² is just 1, so:

x²+ y² = 1
(the equation of the unit circle)

Using Coordinates to Find Trigonometric Functions

The coordinates x and y are two functions of the real variable theta. You can use the coordinates to define the six trigonometric functions of theta.

The point of the unit circle is to make math easier and neater. For example, in the unit circle, you have, for any angle theta, the trig values for sine and cosine are sin(θ) = yand cos(θ) = x. Working from this, you can take the fact that the tangent is defined as being tan(θ) = y/x, and then substitute for x and y to easily prove that the value of tan(θ)also must be equal to the ratio sin(θ)/cos(θ).

Special Points of Interest on the Unit Circle

It is very important to memorize the sine and cosine of the angles created by special right triangles for future use.

 

 

Here is a video on how to easily remember the Unit Circle-

For further help, please watch these videos-
http://www.youtube.com/watch?v=ZffZvSH285c
http://www.youtube.com/watch?v=DIGoK51u0KQ
http://www.youtube.com/watch?v=3GgO7Q_kg8Q

 

 -Kenji Johnston