The unit circle is a circle of radius one placed at the center of the coordinate system. There are many applications of the unit circle in trigonometry, physics and higher level math. This page summarizes some of the many applications of the unit circle.
 Unit Circle Trigonometry
 Unit Circle Physics (Vectors)
 Unit Circle Complex Numbers (Euler’s Formula)
The unit circle visualizes the input and output of the trigonometric functions. As a concept, the unit circle expands the right triangle definitions of the trigonometric functions to all real numbers using the improved circle definitions of the functions as discussed on this page.
The property each trigonometric function corresponds to on the unit circle is shown in the figure above and summarized in the table below. These properties are useful when, for example, converting between polar and cartesian coordinates. They are used throughout the rest of this page.
Geometry  Function 

Sine Function The output of sine corresponds to the distance from a point on the perimeter of the unit circle to the axis. 

Cosine Function The output of cosine corresponds to the distance from a point on the perimeter of the unit circle to the axis. 

Tangent Function The output of tangent corresponds to the length of the line tangent to a point on the unit circle starting from the point and intersecting with the xaxis. 

Cotangent Function The output of cotangent corresponds to the length of the line tangent to a point on the unit circle starting from the point and intersecting with the yaxis. 
The properties of the unit circle are used in physics for decomposing vectors into their components. For example, say we want to calculate the components of a velocity vector given its speed and direction. The trigonometry of the unit circle allows us to decompose the vector into its horizontal velocity using its speed (magnitude) denoted and direction (angle) . This is shown below.
The trigonometry of the unit circle is used in linear algebra for transformation matrices. For example, the matrix above transforms the vector by the angle (theta). Futhermore, because the values are scalar in the range from to , the matrices also inherit some useful properties.
Finally, the unit circle also appears in the plane of complex numbers in the famous formula: Euler’s Formula. Given an angle the formula returns the point on the unit circle in the complex plane corresponding to . This is shown below.
Hopefully this page gives you an idea of some of the applciations of the unit circle. I will keep updating it with examples over time.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
The properties of the unit circle are used in physics for decoming vectors into their components.
Euler's Formula returns the point on the unit circle in the complex plane when given an angle.
To convert a point from the Polar Coordinate System to the Cartesian Coordinate System the functions sine and cosine are used to calculate the x and y component of the corresponding point.
The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.
The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the righttriangle's adjacent side over its hypotenuse.
Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.
Given the angle of a right triangle as input, returns the ratio of the adjacent side over the opposite side.
A complex number is an extension of the real number line where in addition to the "real" part of the number there is a complex part of the number. The properties of complex numbers are useful in applied physics as they elegantly describe rotation.