# Definition:Trigonometric Function

## Definition

There are six basic **trigonometric functions**: sine, cosine, tangent, cotangent, secant, and cosecant.

## Sine

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **sine** of $\angle \theta$ is defined as being $\dfrac {\text {Opposite} } {\text {Hypotenuse} }$.

### Definition from Circle

The sine of an angle in a right triangle can be extended to the full circle as follows:

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $x$-axis.

Then the **sine** of $\theta$ is defined as the length of $AP$.

Hence in the first quadrant, the **sine** is positive.

### Real Numbers

The real function $\sin: \R \to \R$ is defined as:

\(\ds \forall x \in \R: \, \) | \(\ds \sin x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\) |

### Complex Numbers

The complex function $\sin: \C \to \C$ is defined as:

\(\ds \forall z \in \C: \, \) | \(\ds \sin z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots\) |

## Cosine

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cosine** of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $y$-axis.

Then the **cosine** of $\theta$ is defined as the length of $AP$.

Hence in the first quadrant, the **cosine** is positive.

### Real Numbers

The real function $\cos: \R \to \R$ is defined as:

\(\ds \cos x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots\) |

### Complex Numbers

The complex function $\cos: \C \to \C$ is defined as:

\(\ds \cos z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\) |

## Tangent

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **tangent** of $\angle \theta$ is defined as being $\dfrac{\text{Opposite}} {\text{Adjacent}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **tangent** of $\theta$ is defined as the length of $AB$.

Hence in the first quadrant, the **tangent** is positive.

### Real Function

Let $x \in \R$ be a real number.

The real function $\tan x$ is defined as:

- $\tan x = \dfrac {\sin x} {\cos x}$

where:

The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.

### Complex Function

Let $z \in \C$ be a complex number.

The complex function $\tan z$ is defined as:

- $\tan z = \dfrac {\sin z} {\cos z}$

where:

The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.

## Cotangent

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cotangent** of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Opposite}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **cotangent** of $\theta$ is defined as the length of $AB$.

Hence in the first quadrant, the **cotangent** is positive.

### Real Function

Let $x \in \R$ be a real number.

The real function $\cot x$ is defined as:

- $\cot x = \dfrac {\cos x} {\sin x} = \dfrac 1 {\tan x}$

where:

The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.

### Complex Function

Let $z \in \C$ be a complex number.

The complex function $\cot z$ is defined as:

- $\cot z = \dfrac {\cos z} {\sin z} = \dfrac 1 {\tan z}$

where:

The definition is valid for all $z \in \C$ such that $\sin z \ne 0$.

## Secant

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **secant** of $\angle \theta$ is defined as being $\dfrac{\text{Hypotenuse}} {\text{Adjacent}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \tuple {1, 0}$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **secant** of $\theta$ is defined as the length of $OB$.

Hence in the first quadrant, the **secant** is positive.

### Real Function

Let $x \in \R$ be a real number.

The real function $\sec x$ is defined as:

- $\sec x = \dfrac 1 {\cos x}$

where $\cos x$ is the cosine of $x$.

The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.

### Complex Function

Let $z \in \C$ be a complex number.

The complex function $\sec z$ is defined as:

- $\sec z = \dfrac 1 {\cos z}$

where $\cos z$ is the cosine of $z$.

The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.

## Cosecant

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cosecant** of $\angle \theta$ is defined as being $\dfrac {\text {Hypotenuse}} {\text {Opposite}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \tuple {0, 1}$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **cosecant** of $\theta$ is defined as the length of $OB$.

Hence in the first quadrant, the **cosecant** is positive.

### Real Function

Let $x \in \C$ be a real number.

The real function $\csc x$ is defined as:

- $\csc x = \dfrac 1 {\sin x}$

where $\sin x$ is the sine of $x$.

The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.

### Complex Function

Let $z \in \C$ be a complex number.

The complex function $\csc z$ is defined as:

- $\csc z = \dfrac 1 {\sin z}$

where $\sin z$ is the sine of $z$.

The definition is valid for all $z \in \C$ such that $\sin z \ne 0$.

## Also known as

A **trigonometric function** is sometimes referred to as a **direct trigonometric function** so as to distinguish it from an **inverse trigonometric function**.

Some sources use the terms **cyclometric function** or **circular function**.

Some older sources refer to a **trigonometric function** as a **trigonometric ratio**.

## Also see

- Results about
**the trigonometric functions**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**trigonometric function**,**circular function**or**cyclometric function**