definitions in trigonometry

Informal definitions

Given a triangle $ABC$ with a signed angle $x$ at $A$ and a right angle at $B$, the ratios

$$\frac{BC}{AC}\mathit{\hspace{1em}\hspace{1em}}\frac{AB}{AC}\mathit{\hspace{1em}\hspace{1em}}\frac{BC}{AB}$$

are dependent only on the angle $x$, and therefore define functions, denoted by

$$\sin x\mathit{\hspace{1em}\hspace{1em}}\cos x\mathit{\hspace{1em}\hspace{1em}}\tan x$$

respectively, where the names are short for sine, cosine and tangent. Their inverses are rather less important, but also have names:

$\cot x$ $=$ ${\displaystyle \frac{AB}{BC}}={\displaystyle \frac{1}{\tan x}}\text{ (cotangent)}$

$\csc x$ $=$ ${\displaystyle \frac{AC}{BC}}={\displaystyle \frac{1}{\sin x}}\text{ (cosecant)}$

$\sec x$ $=$ ${\displaystyle \frac{AC}{AB}}={\displaystyle \frac{1}{\cos x}}\text{ (secant)}$

From Pythagoras’s theorem we have ${\cos }^{2}x+{\sin }^{2}x=1$ for all (real) $x$. Also it is “clear” from the diagram at left that functions $\cos$ and $\sin$ are periodic with period $2\pi$. However:

Formal definitions

The above definitions are not fully rigorous, because we have not defined the word angle. We will sketch a more rigorous approach.

The power series

$$\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$$

converges uniformly on compact subsets of $ℂ$ and its sum, denoted by $\exp (x)$ or by $e^x$, is therefore an entire function of $x$, called the exponential function. $f(x)=\exp (x)$ is the unique solution of the boundary value problem

$$f(0)=1\mathit{\hspace{1em}\hspace{1em}}{f}^{\prime }(x)=f(x)$$

on $ℝ$. The sine and cosine functions, for real arguments, are defined in terms of $\exp$, simply by

$$\exp (ix)=\cos x+i(\sin x).$$

Thus

$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\mathrm{\cdots }$$

$$\sin x=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\mathrm{\cdots }$$

Although it is not self-evident, $\cos$ and $\sin$ are periodic functions on the real line, and have the same period. That period is denoted by $2\pi$.

Title	definitions in trigonometry
Canonical name	DefinitionsInTrigonometry
Date of creation	2013-03-22 13:55:08
Last modified on	2013-03-22 13:55:08
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	10
Author	Daume (40)
Entry type	Definition
Classification	msc 26A09
Related topic	Trigonometry
Related topic	Sinusoid
Related topic	ComplexSineAndCosine
Related topic	ExampleOnSolvingAFunctionalEquation
Related topic	DerivativesOfSineAndCosine
Related topic	AdditionFormulasForSineAndCosine
Related topic	AdditionFormulaForTangent
Related topic	GoniometricFormulae
Related topic	OsculatingCurve
Defines	sine
Defines	cosine
Defines	exponential
Defines	tangent
Defines	cotangent
Defines	secant
Defines	cosecant
Defines	trigonometric function