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 $\cos z \ne 0$.
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): circular function
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cyclometric function
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): trigonometric function, circular function or cyclometric function
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): direct trigonometric function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): direct trigonometric function