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. xyz_1965

    MHB Is tan(x)^2 proper notation for the trig function tangent squared?

    Is tan^2 (x) the same as tan(x)^2? Note: I could have used any trig function. I know that tan^2 (x) means (tan x)^2. What does tan (x)^2 mean? Is it proper notation?
  2. Leo Liu

    I How did mathematicians discover the expressions of hyperbolic functions?

    The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are $$ \begin{cases} \sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...
  3. T

    What brain exercises improve brain functions based on facts

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  4. anemone

    MHB Expressing First 1000 Positive Integers as Floor Functions

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  5. W

    Elementary Python Questions: Data Frames, k-nary functions

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  6. B

    A Jacobi Elliptic Functions and Integrals

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  7. S

    Correct statement about composite and inverse functions

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  8. samuelfarley

    Using derivatives to determine the increase and decrease of functions

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  9. chwala

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  10. Math Amateur

    MHB Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 .... ....

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  11. Math Amateur

    I Lebesgue Integration of Simple Functions .... Lindstrom, Lemma 7.4.6 ...

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  12. Math Amateur

    MHB Measurable Functions .... Lindstrom, Proposition 7.3.7 .... ....

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  13. Math Amateur

    I Measurable Functions .... Lindstrom, Proposition 7.3.7 .... ....

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  14. J

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  15. B

    Calculus 1 problems: functions, integrals, series

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  16. H

    B What Are the Sign Rules for Second Degree Functions?

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  17. Leo Liu

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  18. N

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  19. T

    MHB Derivatives with Quadratic Functions.

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  20. O

    I Rotate Functions with Derivatives: A Quantum Mechanics Homework

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  21. agnimusayoti

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  22. H

    Find the number of possible functions from one set to another

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  23. E

    B Tangents to a curve as functions of different variables

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  24. benorin

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  25. Physics lover

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  26. Physics lover

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  27. anemone

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  28. E

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  29. L Navarro H

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  30. A

    Medical Text about higher cognitive functions of the brain?

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  31. S

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  32. M

    MHB Show that the two functions are identical

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  33. benorin

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  34. V

    A How to work with expanded functions?

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  35. M

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  36. Mounice

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  37. P

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  38. Avaro667

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  39. T

    Leetcode 728 complier issue with mutliple functions

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  40. C

    I Finite expansion of a fraction of functions

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  41. M

    Taking the derivative of complex functions

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  42. brotherbobby

    Proving that the two given functions are linearly independent

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  43. brotherbobby

    I Proving functions are linearly dependent

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  44. benorin

    I Hypergeometric Functions Identities: n_F_n & (n+1)_F_n

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  45. R

    B A simple doubt on functions.

    Why do we define functions as only as only those graphs which have only one y value for each x value. for eg. we don't say that a circle is a graph of a function,because its graph would have two y values for same x values. what i mean to ask is why not call anything that takes a input and gives...
  46. M

    I Finding Multiple Local Minima of High-Dimensional Functions

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  47. rafgger

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  48. Math Amateur

    I Composition of Two Continuous Functions .... Browder, Proposition 3.12

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  49. Math Amateur

    MHB Composition of Two Continuous Functions .... Browder, Proposition 3.12 .... ....

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  50. F

    Sequence of integrable functions (f_n) conv. to f

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