definitions in trigonometry
Informal definitions
Given a triangle with a signed angle at and a right angle at , the ratios
are dependent only on the angle , and therefore define functions, denoted by
respectively, where the names are short for sine, cosine and tangent. Their inverses are rather less important, but also have names:
From Pythagoras’s theorem we have for all (real) . Also it is “clear” from the diagram at left that functions and are periodic with period . 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
converges uniformly on compact subsets of and its sum, denoted by or by , is therefore an entire function of , called the exponential function. is the unique solution of the boundary value problem
on . The sine and cosine functions, for real arguments, are defined in terms of , simply by
Thus
Although it is not self-evident, and are periodic functions on the real line, and have the same period. That period is denoted by .
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 |