definitions in trigonometry

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

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

\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:

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

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}}^{\infty}\frac{x^{n}}{n!}

converges uniformly on compact subsets of $\mathbb{C}$ 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\qquad f^{{\prime}}(x)=f(x)

on $\mathbb{R}$ . 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!}+\cdots

\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 $2\pi$ .

Defines:

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

Type of Math Object:

Definition

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