Definition:Trigonometric Function

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

Sine

Trigonometry

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

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

Analysis

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

$\displaystyle \sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

$\sin x$ is voiced sine (of) $x$.

Cosine

Trigonometry

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

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

Analysis

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

$\displaystyle \cos x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

$\cos x$ is voiced cosine (of) $x$, or (as written) cos $x$ (pronounced either coss or coz depending on preference).

Tangent

Trigonometry

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

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

Thus it is seen that the tangent is the sine over the cosine.

Analysis

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:

• $\sin x$ is the sine of $x$
• $\cos x$ 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 $\tan z$ is defined as:

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

where:

• $\sin z$ is the sine of $z$
• $\cos z$ is the cosine of $z$.

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

Cotangent

Trigonometry

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

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

Thus it is seen that the cotangent is the reciprocal of the tangent.

It is also seen to be the cosine over the sine.

Analysis

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:

• $\sin x$ is the sine of $x$
• $\cos x$ is the cosine of $x$
• $\tan x$ is the tangent 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 $\cot z$ is defined as:

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

where:

• $\sin z$ is the sine of $z$
• $\cos z$ is the cosine of $z$
• $\tan z$ is the tangent of $z$

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

Secant

Trigonometry

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}} $.

Thus it is seen that the secant is the reciprocal of the cosine.

Analysis

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

Trigonometry

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

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

Thus it is seen that the cosecant is the reciprocal of the sine.

Analysis

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 $\sin z \ne 0$.