Definition:Trigonometric Function

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Definition

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


Sine

Definition from Triangle

SineCosine.png

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 coordinate plane.


SineFirstQuadrant.png


Let $P = \left({x, y}\right)$ 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 $AP$.


Real Numbers

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

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


Complex Numbers

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

\(\displaystyle \sin z\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}\)
\(\displaystyle \) \(=\) \(\displaystyle z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots\)


Cosine

Definition from Triangle

SineCosine.png

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 coordinate plane.


CosineFirstQuadrant.png


Let $P = \left({x, y}\right)$ 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 $AP$.


Real Numbers

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

\(\displaystyle \cos x\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }\)
\(\displaystyle \) \(=\) \(\displaystyle 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots\)


Complex Numbers

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

\(\displaystyle \cos z\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\)


Tangent

Definition from Triangle

SineCosine.png

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 coordinate plane.


TangentFirstQuadrant.png


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 $A = \left({1, 0}\right)$.

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


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


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

Definition from Triangle

SineCosine.png

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 coordinate plane.


CotangentFirstQuadrant.png


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 $A = \left({0, 1}\right)$.

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


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


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

Definition from Triangle

SineCosine.png

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 coordinate plane.


SecantFirstQuadrant.png


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 $A = \left({1, 0}\right)$.

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


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


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

SineCosine.png

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 coordinate plane.


CosecantFirstQuadrant.png


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 $A = \left({0, 1}\right)$.

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


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


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.


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