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. Math Amateur

    MHB HIGHLY Rigorous Treatment of the Trigonometric Functions

    I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ... Can someone please suggest (i) an online...
  2. T

    Evaluate the integral (inverse trig functions)

    Homework Statement [23/4, 2] 4/(x√(x4-4)) Homework Equations ∫ du/(u√(u2 - a2)) = 1/a(sec-1(u/a) + c The Attempt at a Solution I first multiplied the whole thing by x/x. This made the problem: 4x/(x2√(x4 - 4)) Then I did a u substitution making u = x2. Therefore, du = 2xdx. I multiplied by...
  3. G

    Large n-pt functions renormalized by small n-pt functions?

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  4. S

    Exploring Solutions to a Functional Equation with Real Variables

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  5. Philethan

    Is the limit of functions necessarily equal to "itself"?

    As I read in the James Stewart's Calculus 7th edition, he said: My question is: Is f(x)\rightarrow 0 the same as f(x) = L? For example, f(x) = x^2 \displaystyle\lim_{x\rightarrow 5}f(x) = 25 I can say that f(x) = x^2 approaches 25 as x approaches 5. Therefore, can I say that the...
  6. Math Amateur

    MHB How Do You Correctly Format Limits and Derivatives in LaTeX?

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  7. Robsta

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  8. N

    Link between harmonic functions and harmonic oscillators?

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  9. evinda

    MHB What is the Time Complexity of These Binary Search Tree Functions?

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  10. J

    What is the output of two cosine functions?

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  11. evinda

    MHB Cardinality of continuous real functions

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  12. S

    Complex number problem with trig functions

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  13. L

    Finding constants in exponential functions

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  14. S

    Help with the inverse of some functions

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  15. J

    TI89 calculator solver in functions?

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  16. Robsta

    Gram-Schmidt Orthonormal Functions

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  17. Mr Davis 97

    Defining differentitation and integration on functions

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  18. L

    Shifting and inverse functions

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  19. TheDemx27

    Graphing Functions in n Dimensions, Parametric Equations

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  20. karush

    MHB Integrating a Product of Trig Functions

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  21. N

    For f(x) = abs(x^3 - 9x), does f'(0) exist

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  22. ?

    Can a Function be Both Even and Odd at the Same Time?

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  23. 22990atinesh

    Counting One-to-One Functions from n to m with Property f(i)<f(j)

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  24. K

    Deducing Basis of Set T from Coordinates in Matrix A with Respect to Basis S

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  25. I

    MHB Definitions of Functions and Spaces

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  26. G

    Functions of local operators

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  27. vktsn0303

    Why are transcendental functions called so?

    I have learned that they are called so because they cannot be expressed with the help of elemental methods of mathematics such as addition, subtraction, multiplication and division. But then isn't the whole of mathematics itself based on the elemental methods?
  28. angeli

    Application of quadratic functions to volleyball

    Homework Statement A player hits a volleyball when it is 4 ft above the ground with an initial vertical velocity of 20 ft/s (equation would be h = -16t2 + 20t + 4). What is the maximum height of the ball? Homework Equations quadratic formula The Attempt at a Solution t = -20 ±√202 - 4(-16)(4)...
  29. angeli

    Application of quadratic functions to volleyball

    hi! i don't quite know how to start solving for this. i understand the problem and what it's asking for but i have no idea how to start solving for it. In a volleyball game, a player from one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball...
  30. R

    Finding Solutions to a Step Function Integral

    Homework Statement This is from Apostol's Calculus Vol. 1. Exercise 1.15, problem 6.(c) Find all x>0 for which the integral of [t]2 dt from 0 to x = 2(x-1) Homework Equations [t] represents the greatest integer function of t. The Attempt at a Solution [/B] Integral of [t]2 dt from 0 to x...
  31. S

    Generalization of combinatorial generating functions?

    Generating functions defined in terms of algebraic operations on real valued variables are used to enumerate answers to certain combinatorial problems. ( This morning, the exposition http://www.cs.cornell.edu/courses/cs485/2006sp/lecture%20notes/lecture11.pdf is the first of many hits on the...
  32. M

    Integrals and gamma functions manipulation

    Homework Statement I am working through some maths to deepen my understanding of a topic we have learned about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on...
  33. ellipsis

    Covering space of implicit vs parametric functions

    Hello PF, I've got a curiosity question someone may be able to indulge me on: The set of implicit functions covers a certain function-space - the set of all functions that can be represented by an implicit relation. Parametric functions also covers a function-space, that at least overlaps...
  34. B

    MHB Advanced Calculus - Continuous Functions

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  35. I

    C/C++ Exploring Recursive Functions: Benefits & Uses

    Hi :o Recursion. Recursive functions. What are they used for and how they helpful?
  36. I

    C/C++ C++ String Functions with Pointers

    Assign the first instance of The in movieTitle to movieResult. Sample program: #include <iostream> #include <cstring> using namespace std; int main() { char movieTitle[100] = "The Lion King"; char* movieResult = 0; <STUDENT CODE> cout << "Movie title contains The? "; if...
  37. K

    MHB Finding Formula without using any trig functions

    Find a formula for g(x)= sin(arccos(4x-1)) without using any trigonometric functions. I have the answer key right in front of me, but i still get how to start it off or the steps in solving these kind of questions or how to do it at all :/ Thanks!
  38. 4

    Why is a set of functions v(t) dense in L^2

    Hello, I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...
  39. M

    Even and Odd functions - Fourier Series

    Hello everyone, I know that the integral of an odd function over a symmetric interval is 0, but there's something that's bothering my mind about it. Consider, for example, the following isosceles trapezoidal wave in the interval [0,L]: When expressed in Fourier series, the coefficient...
  40. Avatrin

    Ideal gasses, bonds and partition functions

    Hi I am struggling immensely with understand some aspects of chemical thermodynamics: 1) Let's say I have a solid with N atoms and am examining the ionization of individual atoms, and I am supposed to think of the electrons as ideal gasses. Or, 2) a solid or liquid is in thermal equilibrium...
  41. B

    Composing Two Complex Functions

    Homework Statement Suppose we have the function ##f(z) = x + iy^2## and a contour given by ##z(t) = e^t + it## on ##a \le t \le b##. Find ##x(t)##, ##y(t)##, and ##f(z(t))##. Homework EquationsThe Attempt at a Solution Well, ##x(t)## and ##y(t)## are rather simple to identity. However, I am...
  42. B

    Pre-Calc. Question: Graphing Rational Functions

    When you have a rational function, such as: 3x-5/x-1 After attaining things like the x and y intercepts and asymptotes, how do you know how many "pieces" of the graph there are? With linear functions/equations, you know it's a single line. Even quadratic graphs are a single piece - albeit...
  43. T

    MHB Finding area between 2 functions

    1. I have to find the area between $x = 2y^2$ and $x = 1 - y$ I find the intersection points $ 1 -y = 2y^2$ $2y^2 + y - 1= 0 $ $(2y - 1)(y + 1)= 0$ so y = 1 and -1 However, x = y - 1 is not a vertical line so I am not sure how 1 and -1 can be intersections. Also, when I plug these...
  44. T

    MHB Finding Area between 3 functions

    I need to find the area bounded by: $y = \sqrt{x}$, $y = x/2$, and $x = 9$. I found that the intersecting point is 4 and $y = \sqrt{x}$ is the smaller function between 4 and 9 so: $$\int_{4}^{9}\frac{x}{2} - \sqrt{x} \,dx$$ and I get $$ \left[ \frac{x^2}{4} - \frac{2x^{3/2}}{3}\right]_4^9...
  45. T

    MHB Finding Area between 2 functions

    Hi, I have this problem to find the area between 2 curves: $y = x^2$ and $y = \frac{2}{x^2 +1}$ I found that the points of intersection are -1 and 1 and it is symmetrical. I get $2\int_{0}^{1} \ \frac{1}{x^2 + 1} - x^2 dx$which I am unable to solve. I have tried u-substitution but I end up...
  46. T

    MHB What is the area between the functions $y = |2x|$ and $y = x^2 - 3$?

    Hi, I need to find the area between these 2 functions: $$y = |2x|$$ and $$y = x^2 - 3$$ So I need to find the points of intersection: $$|2x| - x^2 + 3 = 0$$ for which I get x = 3, -1 However, since there are no negative x values in y = |2x| I get $x = 3, 1$ I find that $y = |2x| $is...
  47. T

    MHB So, the area between the two functions is 72 units squared.

    I need to find the are between $$y1 = 12 - x^2$$ and $$ y2 = x^2 - 6$$. Since y1 is greater, I subtract y2 from y1 getting: $$ \int 18 - 2x^2$$ which is $$18x - 2x^3 / 3$$, The intersecting points are $$x = -3 and x= 3$$. So I find $$18x - 2x^3 / 3 from x = 3 to x = -3$$(I'm trying to...
  48. 2

    Subtracting functions on specified domains

    OK, so I posted this a few days ago: https://www.physicsforums.com/threads/subtracting-the-overlap-of-functions.784184/#post-4925108 What I've come to discover is that I want to understand how I can subtract f(x) on domain [b,c] from g(x) on domain [a,d]. I want to be able to disregard both...
  49. Fosheimdet

    Harmonic Functions: Laplace's Equations & Analytic Functions

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  50. 2

    Subtracting the overlap of functions

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