definitions in trigonometry

Informal definitions

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


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

sinx  cosx  tanx

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

cotx = ABBC=1tanx (cotangent)
cscx = ACBC=1sinx (cosecant)
secx = ACAB=1cosx (secant)

From Pythagoras’s theorem we have cos2x+sin2x=1 for all (real) x. Also it is “clear” from the diagram at left that functions cos and sin are periodic with period 2π. 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 seriesMathworldPlanetmath


converges uniformly on compact subsets of and its sum, denoted by exp(x) or by ex, is therefore an entire functionMathworldPlanetmath of x, called the exponential functionDlmfDlmfMathworldPlanetmath. f(x)=exp(x) is the unique solution of the boundary value problem

f(0)=1  f(x)=f(x)

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




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

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 functionDlmfMathworld