# Trigonometry Index

Trigonometry is the study of triangles. The subject is a subset of geometry and focuses on the properties of triangles, especially that of the right triangle. This page lists pages on this site tagged with the subject of trigonometry. For an introduction to the study of trigonometry see here.

## Notation

Angle | Notation

The notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.

Line | Notation

A line is denoted by two letters representing the start and end of the line with a line over top.

Perpendicular Angle | Notation

A perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to π/2 radians or 90°.

Perpendicular Lines | Notation

The symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.

Triangle | Notation

A triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.

## Concepts

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

30 60 90 Triangle | Concept

The triangle defined by the three angles: 30 degrees, 60 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.

45 45 90 Triangle | Concept

The triangle defined by the three angles: 45 degrees, 45 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.

Angle | Concept

An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.

Cartesian Coordinate System | Concept

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.

Degrees | Concept

Degrees is a unit of measure for angles. A full rotation is equal to 360 degrees. In the cartesian coordinate system, degrees are measured starting from the rightmost edge of the circle.

Difference of Two Angles Identities | Concept

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

Double Angle Identities | Concept

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

Half Angle Identities | Concept

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

Inscribed Angle Theorem | Concept

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

Law of Cosines | Concept

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

Law of Sines | Concept

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.

Pi | Concept

The greek letter π (pi) is a naturally occurring number that is defined by any circle's circumference divided by its diameter.

Polar Coordinate System | Concept

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

Ptolemy's Theorem | Concept

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

Pythagorean Identity | Concept

The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean's theorem and the properties of the unit circle.

Pythagorean Theorem | Concept

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

Radians are a unit that measure 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.

Right Triangle | Concept

A right triangle is a triangle where one of the three angles is a perpendicular angle. There are three sides of the right triangle: the adjacent, opposite, and hypotenuse sides.

Similar Triangles | Concept

Similar Triangles are two triangles that share the same three angles making them proportional to each other.

Special Triangle | Concept

There are two special right triangles in geometry defined by their three angles the 45 45 90 degrees and the 30 60 90 degrees.

Sum of Two Angles Identities | Concept

The sum of two angles identities express the cosine and sine of the sum of two angles in terms of their individual cosine and sine components.

Tau | Concept

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

Triangle | Concept

A triangle is a three sided geometric shape. The shape forms a basis for the subject of trigonometry and is used throughout mathematics and programming.

Trigonometric Functions | Concept

There are six total 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 | Concept

The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.

Unit Circle Chart | Concept

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

The unit circle uses the radians unit system which measures angles the radius of the circle. One radian is equal the arc-length required to travel the length of the radius of the circle.

Unit Circle Table | Concept

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.

Unit Circle Trigonometry | Concept

The unit circle demonstrates the output of the trigonometric functions sine, cosine and tangent through the shared geometry of the right-triangle defined by a point on the perimeter of the circle.

## Examples

Circumference of Circle Radius 1 | Example

To calculate the circumference of a circle multiply the radius by the constant τ (tau) to get the result.

Circumference of Circle Radius 2 | Example

To calculate the circumference of a circle multiply the radius by the constant τ (tau) to get the result.

Convert Angle From Degrees to Radians | Example

To convert degrees to radians, multiply the angle by pi and then divide by 180.

Convert Angle From Radians to Degrees | Example

To convert an angle from radians to degrees multiply by the common ratio of 360° divided by τ (6.283) radians.

Convert Cartesian to Polar Coordinates | Example

To convert a point from the Cartesian Coordinate System to the Polar Coordinate System the formula for the distance between two points and the arctangent function are used to calculate the radius and angle of the correponding point.

Convert Polar to Cartesian Coordinates | Example

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.

Derive Difference of Two Angles Identities | Example

To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.

Derive Double Angles Identities (Algebra) | Example

The double angle identities can be derived using the sum of two angle identities in combination with Pythagorean's identity.

Derive Double Angles Identities (Inscribed Angle) | Example

The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.

Derive Half Angle Identities (Algebra) | Example

The half-angle identities can be derived from the double angle identities by transforming the angles using algebra and then solving for the half-angle expression.

Derive Law of Sines | Example

To derive the equation for the law of sines, observe the shared perpendicular line by two angles. The third angle can be included by repeating the process.

Derive Law of Sines (Inscribed Triangle) | Example

The law of sines can be derived using a triangle inscribed on the perimeter of a circle. The proof uses the inscribed angle theorem.

Derive Sum of Two Angles Addition Formula | Example

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

Derive Sum of Two Angles Identities | Example

To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.

Pythagorean Theorem 3 4 5 Right Triangle | Example

This example shows how the pythagorean theorem is true for a right triangle with sides length 3, 4, and 5.

## Functions

Sine | Function

The sine function returns the sine of a number provided in radians. In geometric terms, given the angle of a right-triangle as input, the function returns the ratio of the triangle's opposite side over its hypotenuse.

Cosine | Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

Tangent | Function

Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.

Arc Sine | Function

Given a number representing the the ratio of a right triangle's opposite side over its hypotenuse returns the corresponding angle.

Arc Cosine | Function

Given a number representing the the ratio of a right triangle's adjacent side over its hypotenuse returns the corresponding angle.

Arc Tangent | Function

Given a number representing the the ratio of a right triangle's opposite side over its adjecent side returns the corresponding angle.

Arc Tangent 2 | Function

Given a number representing the the ratio of a right triangle's opposite side over its adjecent side returns the corresponding angle.

Cosecant | Function

Given the angle of a right triangle as input, returns the ratio of the hypotenuse over the opposite side. The cosecant function is the inverse of the sine function.

Cotangent | Function

Given the angle of a right triangle as input, returns the ratio of the adjecent side over the opposite side.

Secant | Function

Given the angle of a right triangle as input, returns the ratio of the hypotenuse over the adjacent side. The secant function is the inverse of the cosine function.

## Graphs

Arc Cosine | Graph

The graph of the inverse cosine or arc-cosine function from -1 to 1.

Arc Sine | Graph

Graph of the cos(x) from 0 to 2 PI. Cosine is a trigonometric function that when given the angle of a right triangle, returns the ratio of its adjacent side over the hypotenuse.

Arc Tangent | Graph

Graph of the cos(x) from 0 to 2 PI. Cosine is a trigonometric function that when given the angle of a right triangle, returns the ratio of its adjacent side over the hypotenuse.

Cosine | Graph

Graph of the cos(x) from 0 to 2 PI. Cosine is a trigonometric function that when given the angle of a right triangle, returns the ratio of its adjacent side over the hypotenuse.

Sine | Graph

The graph of SIN(θ) from 0 to τ. Sine is a trigonometric function that returns the ratio of a right-triangle's opposite side over its hypotenuse.

Tangent | Graph

A graph of tan(x) from 0 to 2 PI. Tangent is a trigonometric function that when given the angle of a right triangle, returns the ratio of the opposite side over the adjacent side.

Secant | Graph

Graph of the sec(x) from 0 to 2 PI. Secant is a trigonometric function and is the inverse of the cosine function.

Cosecant | Graph

Graph of csc(x) from 0 to 2 PI. Cosecant is a trigonometric function related to the geometry of the right triangle. It is the inverse of the sine function.

Cotangent | Graph

Graph of the cot(x) from 0 to 2 PI. Cotangent is a trigonometric function and is the inverse of the tangent function.

## Interactives

Unit Circle | Interactive

This interactive demonstrates the connection between the unit circle and the trigonometric functions sine, cosine and tangent.

Cosine Function | Interactive

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.

Sine Function | Interactive

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.