Trigonometric Functions

The trigonometric functions relate the geometry of the right triangle to its angle. There are three main functions: cosine function, sine function, and tangent function. Each function takes the angle of a right triangle as input and returns the ratio of two of its sides. Shown below are the definitions of the main circle functions:

Right Triangle

Trig. Functions Introduction

The functions can be grouped in three groups: 1) the main functions: sine, cosine and tangent. 2) The reciprocal functions: cosecant, secant and cotangent. 3) The inverse or arc functions: arc-sine, arc-cosine and arc-tangent.

Sine Function
sin(θ)
Cosine Function
cos(θ)
Tangent Function
tan(θ)
Cosecant Function
csc(θ)
Secant Function
sec(θ)
Cotangent Fuction
cot(θ)
Arc Sine Function
arcsin(n)
Arc Cosine Function
arccos(n)
Arc Tangent Function
arctan(n)
Arc Tangent 2 Function
arctan(x,y)

Main Functions

The main functions: sine, cosine, and tangent are the basis of trigonometry. They are used throughout physics and mathematics to describe sinusoidal waveforms, circles and much more. Calculators often have the option to toggle between the radians and degrees angle systems, but some programming languages and calculators only offer the functions using radians.

Sine Function
sin(θ)
Cosine Function
cos(θ)
Tangent Function
tan(θ)

Sine Function

Right Triangle

Given an angle provided in radians the sine function returns the ratio of the opposite side over its hypotenuse. For example, given the angle of radians equivalent to the sine function returns the ratio .

Since the function returns a ratio, the relative size of the right-triangle does not matter, only its measured angle. All possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle. Click and drag either of the blue control points in the interactive below to see the input and output of the function.

Cosine Function

Right Triangle

Given an angle provided in radians the cosine function returns the ratio of the adjacent side over its hypotenuse. For example, given the angle of radians equivalent to the cosine function returns the ratio .

Since the function returns a ratio, the relative size of the right-triangle does not matter, only its measured angle. All possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle. Click and drag either of the blue control points in the interactive below to see the input and output of the function.

Tangent Function

Right Triangle

Given an angle the tangent function returns the ratio of the opposite side over its adjacent side.

Graph of the tangent function

Reciprocal Functions

The three functions secant, cosecant, and cotangent are the reciprocal functions of cosine, sine, and tangent. Just like the main functions, these functions take in an angle of a right-triangle as input and return the ratio of two of its sides. Each function can be expressed as the reciprocal of its counter-part. Note, the functions cosecant and secant have a partly historical context and are not as common to encounter.

Cosecant Function
csc(θ)
Secant Function
sec(θ)
Cotangent Fuction
cot(θ)

Cosecant Function

The cosecant function returns the reciprocal of the sine function. This is visualized in the graph below where the sine function is highlighted in green.

Graph of the cosecant function Graph of the cosecant and sine function

Secant Function

The secant function returns the reciprocal of the cosine function. This is visualized in the graph below where the cosine function is highlighted in green.

Graph of the secant function Graph of the secant and cosine function

Cotangent Function

The cotangent function returns the reciprocal of the tangent function. This is visualized in the graph below.

Graph of the cotangent function Graph of the cotangent and tangent function

Arc Functions

The arc-functions are the inverse of cosine, sine, and tangent. Given the ratio of two sides of the right-triangle the functions return the corresponding angle.

Arc Cosine Function
arccos(n)
Arc Sine Function
arcsin(n)
Arc Tangent Function
arctan(n)
Arc Tangent 2 Function
arctan(x,y)

Arc Cosine Function

Graph of the arc cosine function Graph of the cosine function

Arc Sine Function

Graph of the arc sine function Graph of the sine function

Arc Tangent Function

Graph of the arc tangent function Graph of the tangent function

Unit Circle

Since the concept of an angle is closely linked the circle, the trigonometric functions are also refered to as the circular functions and are studied using the unit circle. This relationship between the functions and the circle is also demonstrated by the fact that the functions are periodic, meaning they repeat over and over, so they are used in applications to model cyclic patterns. There is also a set of trigonometric functions called the hyperbolic functions that relate to the shape of the hyperbola.

The circular trigonometric functions can be visualized by placing the right triangle at the center of the coordinate system. The sides of the right-triangle are exchanged the corresponding variable in the coordinate systems which correspond to the point the triangle forms.

Right Triangle
Right Triangle in Cartesian Coordinate System

The circle the functions relate to is a circle of radius also placed at the center of the coordinate system. The significance of the is because the functions return a ratio of sides, the geometry of the unit circle demonstrates all possible output of the functions.

Right Triangle in Cartesian Coordinate System

Now, we can represent the six main functions in terms of , and . Each function can be geometrically related to the right triangle and a line drawn tangent to the point on the arc of the circle.

Trigonometric Functions

Good Links

Hyperbolic Functions | Concept

The hyperbolic functions are a set of trigonometric functions that relate the geometry of a right triangle to its hyperbolic angle.

Trigonometric Identities | Concept

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

Trigonometry | Concept

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.