PlanetMath: definitions in trigonometry

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definitions in trigonometry

(Definition)

$\includegraphics{trig.eps}$

Informal definitions

Given a triangle with a signed angle at and a right angle at , the ratios

$\displaystyle \frac{BC}{AC}\qquad \frac{AB}{AC}\qquad \frac{BC}{AB}$

are dependent only on the angle

, and therefore define functions, denoted by

$\displaystyle \sin x\qquad \cos x\qquad \tan x$

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

$\displaystyle \cot x$	$\displaystyle =$	$\displaystyle \frac{AB}{BC} = \frac{1}{\tan x}$ (cotangent)
$\displaystyle \csc x$	$\displaystyle =$	$\displaystyle \frac{AC}{BC} = \frac{1}{\sin x}$ (cosecant)
$\displaystyle \sec x$	$\displaystyle =$	$\displaystyle \frac{AC}{AB} = \frac{1}{\cos x}$ (secant)

From Pythagoras's theorem we have $\cos^2 x+\sin^2 x = 1$ for all (real)

. 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

$\displaystyle \sum_{n=0}^\infty\frac{x^n}{n!}$

converges uniformly on compact subsets of $\mathbb{C}$ and its sum, denoted by $\exp(x)$ or by

, is therefore an entire function of

, called the exponential function. $f(x)=\exp(x)$ is the unique solution of the boundary value problem

$\displaystyle f(0)=1\qquad f'(x)=f(x)$

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

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

Thus

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

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

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 $2\pi$ .

"definitions in trigonometry" is owned by Daume. [ full author list (2) | owner history (1) ]

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Also defines:

sine, cosine, exponential, tangent, cotangent, secant, cosecant, trigonometric function

Attachments:: Prosthaphaeresis formulas (Proof) by mathfanatic
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sohcahtoa (Definition) by Wkbj79
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rigorous definition of trigonometric functions (Derivation) by rspuzio
rigorous definition of tangent function (Derivation) by rspuzio
derivatives of $\sin x$ and $\cos x$ (Theorem) by Wkbj79

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Cross-references: line, periodic functions, arguments, boundary, solution, exponential function, entire function, sum, compact subsets, converges uniformly, power series, period, periodic, real, Pythagoras' theorem, functions, right angle, angle, triangle, definitions
There are 72 references to this entry.

This is version 7 of definitions in trigonometry, born on 2003-08-30, modified 2007-10-30.
Object id is 4676, canonical name is DefinitionsInTrigonometry.
Accessed 40112 times total.

Classification:

AMS MSC:

26A09 (Real functions :: Functions of one variable :: Elementary functions)

Pending Errata and Addenda

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