# Concepts

A collection of numbers, theorems, identities, and systems from mathematics.

The binomial theorem shows a generalized way to expand the sum of two variables raised to some power n.

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Geometric construction is a classic form of math that studies building forms and shapes using a compass and straight edge.

Complex numbers are an extension of the real number system with useful properties that model two dimensional space and trigonometry.

The cotangent identity expresses cotangent as the reciprocal of the tangent function.

Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.

The difference of two angles identities expresses the cosine and sine of the difference of the two angles in terms of their individual components.

The double angle identities give the sine and cosine of a double angle in terms of the sine and cosine of a single angle.

Euler's number is a naturally occurring number related to exponential growth and exponential decay. It is also shorthand for the exponential function.

A function takes input and produces output. The idea is a useful way to abstract away complexity and, especially in the age of computers, is a practical tool to solve problems.

The fundamental theory of arithmetic states that every number greater than 1 is either a prime number or composed by a unique product of prime numbers.

The fundamental theorem of calculus relates integration to differentiation by defining the integral of a continuous function on a closed bounded interval.

The half angle identities give the sine and cosine of a half angle in terms of the sine and cosine of an angle.

The inscribed angle theorem states that an inscribed angle in a circle is equal to one-half the central angle.

The law of cosines is a more general form of the Pythagorean theorem that relates the squares of the sides together using the cosine function.

The law of sines is an equation that relates the three sides of a triangle with the three angles of a triangle using the sine function.

The mean value theorem states that for a function f that is continuous on the closed interval [a,b] and differentiable on the interval (a,b) there exists some point c where a < c < b whose slope is equal to the slope formed by the points a and b.

The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the normal distribution forms a "bell curve".

The order of operations is an agreed-upon standard that describes what operations should be performed first in a math expression.

The Polar Coordinate System describes points in space using an angle and radius relative to the origin.

The prime factorization of a number decomposes the number into a unique sequence of prime factors.

A prime number is a number that is only divisible by itself and one. The set of prime numbers is infinitely big. The first prime numbers are 2, 3, 5, 7, 11, ... and continues on forever.

Ptolemy's Theorem relates the diagonal length of an inscribed quadrilateral to the lengths of its sides.

The Pythagorean Identity is a fundamental relation in trigonometry relating the square of the sine and cosine functions of an angle. It is derived from the Pythagorean theorem when applied to the unit circle.

The Pythagorean theorem equates the square of the sides of a right triangle together.

Radians are a unit that measures angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

This page compares and contrasts the two systems of measuring angles in math: radians and degrees, and explains why radians is the preferred unit of measure for angles.

The reciprocal identities define three trigonometric functions: cosecant, secant and cotangent in terms of the primary trigonometric functions: sine, cosine and tangent.

The standard normal distribution is the special case of a normal distribution with a mean of 0 and a standard deviation of 1. The distribution has historical significance because it allows standardized values to be referenced in a look-up table rather than calculated by hand.

The sum of two angles identities are trigonometric identities that express the cosine and sine of the sum of two angles in terms of the trigonometry of the individual angles.

The tangent identity expresses the tangent of an angle in terms of cosine and sine.

A Taylor series is a tool in mathematics to define a function in terms of an infinite power series.

There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

Trigonometric identities are a set of equations derived from the properties of triangles and circles. The identities are used to transform and manipulate math expressions.

The unit circle is a circle of radius one placed at the origin of the coordinate system. A point's geometry on the unit circle corresponds to the input and the output of the trigonometric functions.

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into equal parts.

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into equal parts.

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into equal parts.

The unit circle table contains values for the points along the unit circle. Each point is described by an angle and corresponds with a x and y component.

The Greek letter π (pi) is a geometric constant approximately equal to 3.1416. Its value is equal to the length of any circle's circumference divided by its diameter.

The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.