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.

Degrees | Concept

Degrees are a unit that measures angle of rotation as a fraction of the number 360.

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.

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 is an equation that equates the square of the sides of a right triangle together.

Radians | Concept

Radians are a unit that measure angles 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 double angle identity...

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.

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.

Examples

Angle of Right Triangle Given Two Sides | Example

To calculate the the angle of a right triangle you can use the inverse trigonometric functions arcsin, arccosine, and arctang.

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 Point From Polar Coordinates to Cartesian Coordinates | Example

To convert a point from the Polar Coordinate System to the Cartesian Coordinate System you can use the definition of sine and cosine to solve for the x and y component of the corresponding point.

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.

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

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

Cosine | Function

Given the angle of a right triangle as input, returns the ratio of the 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.

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.

Interactives

Unit Circle | Interactive

This page displays the connection between the unit circle and the trigonometric functions sine and cosine.

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.