# Interactives

This is a collection of interactive graphics. Each interactive has controls which allow the graphic to be interfaced with manipulated.

## Degrees

This interactive illustrates the degree angle system where a full rotation is equal to .

## Radians

This interactive illustrates the radian angle system where a full rotation is equal to (tau) radians.

## Prime Factorization

This interactive generates the prime factorization tree for the given input number. The tree demonstrates how a number can be decomposed into a unique sequence of prime numbers (highlighted in green). This sequence of numbers is called the prime factorization of a number and is related to the Fundamental Theorem of Arithmetic. Each sequence is unique, meaning that no two numbers will have the same prime factorization. The product of the prime factorization of a number will always be equal to the number.

## Unit Circle

This interactive illustrates the connection between the unit circle and the trigonometric functions cosine, sine and tangent. Click and drag the two control points to change the angle or press the animate button. The symbol represents the current angle corresponding to the right triangle with an adjacent side represented by and an opposite side represented by . Note, by convention angles are measured using radians.

## Modular Arithmetic Wheel

The modular arithmetic wheel is a wheel of numbers starting from 0 and counting upwards: 0, 1, 2, 3, and so on. The wheel is divided into a number of sections that is controlled by the slider. The wheel is “modular” since the number of sections corresponds to modular arithmetic and visually represents “modulo ” where is the number of sections. The individual numbers in the wheel can be selected to demonstrate the result of .

## Exponential Trees

This interactive demonstrates the basic properties of exponential growth using the structure of a tree. The base of the exponent corresponds to the number of branches that sprout at each intersection or “branching factor” of the tree. The exponent corresponds to the number of levels in the tree. This is expressed mathematically as some number *b* (base) raised to the *y* power (exponent).

## Polar Coordinate System

This interactive demonstrates the Polar Coordinate System. In the polar coordinates the position of a point is defined by its distance from the origin (radius) and its angle relative to the origin. The angle is measured in radians where one full rotation is equal to Tau (τ) radians, where Tau is a geometric constant approximately equal to 6.28.

## Cartesian Coordinate System

This interactive demonstrates the Cartesian Coordinate System. Every point’s position is defined by two components: the x-component which measures the horizontal distance from the origin and the y-component which measures the vertical distance from the origin.

## Cosine Function

This interactive demonstrates the connection between a right triangle of hypotenuse one and the graph of the cosine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box. main: unit-circle-cosine It is important to note that the adjacent side of the triangle corresponds to the x component of the point. This is because the circle’s center is at the point (0,0), separate from the graph.

## Sine Function

This interactive demonstrates the connection between every right triangle of hypotenuse one and the graph of the sine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box.

## Angle

This interactive illustrates the definition of an angle formed by two rays. The point where the two rays meet is called the vertex of the angle. Angles are measured using two unit of measure in math: radians and degrees.

## Angle Between Two Vectors

This interactive demonstrates the angle formed between two vectors.

## Area Between Two Vectors

This interactive demonstrates how to vectors form the shape of a parallelogram. This area of this shape is of particular interest because of its application to computer graphics, approximating the surface area of a shape, and more. To calculate the area, we can use the property of the cross product operator between to vectors: the magnitude of the cross product of two vectors is equal to area between them.

## Line Defined By Two Points

This interactive demonstrates how to points uniquely define a line.

## Point Where Two Lines Intersect

This interactive demonstrates the point where two lines intersect. The interactive has four blue control points which define the shape of two lines. Click and drag any of the control points to see the point of intersection change. Calculating the Position of the Point The general solution to finding the point of intersection is shown below for the two lines given by the slope intercept equations below. Steps Start by setting the equation of the two lines equal to each other.