What is Functions: Definition and 1000 Discussions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

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  1. C

    Comp Sci Book Database Implementation C++

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  2. R

    Exploring Oscillatory Behavior of Sinusoidal Functions

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  3. redtree

    B The expectation value of superimposed probability functions

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  4. M

    MHB Compositions, Inverses and Combinations of Functions

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  5. Math Amateur

    MHB Continuous Functions on Intervals .... B&S Theorem 5.3.2 ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...
  6. M

    MHB Trigonometry and periodic functions

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  7. Math Amateur

    MHB Continuous Functions - Thomae's Function ....

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  8. Math Amateur

    MHB Limits of Functions .... B&S Theorem 4.2.9 .... ....

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  9. Math Amateur

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  10. Mr Davis 97

    Functions and Analysis with a fixed-point

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  11. B

    Solutions to Equations Involving Exponential and Trig Functions

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  12. karush

    MHB 15.1.55 Find the mass of the plates with the following density functions

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  13. S

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  14. L

    I Can a Circular Function with Complex Variable Represent a 3D Graph?

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  15. Avatrin

    I Rigorously understanding chain rule for sum of functions

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  16. ecoo

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  17. Oats

    I Must functions really have interval domains for derivatives?

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  18. P

    A If [A,B]=0, are they both functions of some other operator?

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  19. binbagsss

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  20. M

    What is Implicit Differentiation for a Circle?

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  21. evinda

    MHB All boolean functions are computed by a depth 2 circuit

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  22. T

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  23. binbagsss

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  24. Telemachus

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  25. J

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  26. Drakkith

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  27. T

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  28. D

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  29. Y

    MHB Partial Order Relation on a Functions Set

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  30. A

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  31. S

    I Difference between holomoprhic and analytic functions

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  32. R

    Integral simplification using Bessel functions

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  33. maistral

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  34. H

    MHB Partial Derivatives of Functions

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  35. Wrichik Basu

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  36. G

    B Add two functions, same frequency to produce one greater?

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  37. V

    A Game Theory: Are the payoff functions πi continuous?

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  38. SherLOCKed

    A Operation of Hamiltonian roots on wave functions

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  39. Bunny-chan

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  40. Mr Davis 97

    I Continuity of composition of continuous functions

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  41. awholenumber

    B Some doubts about functions.... (changing the independent variable from time to position)

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  42. MAGNIBORO

    I Why are there no other gamma functions?

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  43. F

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  44. V

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  45. V

    MHB Can We Prove a Function with Intermediate Value Property is Continuous at x?

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  46. maistral

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  47. S

    A Gauge-invariant operators in correlation functions

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  48. binbagsss

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  49. G

    Expectation values as a phase space average of Wigner functions

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  50. binbagsss

    Elliptic functions, removable singularity, limits,

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