# 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 $2\pi$.

 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