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.

View More On Wikipedia.org
Recent contents Top users
  1. T

    Probability Generating Functions Question

    Homework Statement In playing a certain game, your ability scores are determined by six independent rolls of three dice. After each set of six rolls, you are given the choice of keeping your scores or starting over. (a) How many times should you expect to start over in order to get a set of...
  2. C

    Finding partition functions of statistical system

    Homework Statement Consider a zipper of N links, each of which can either be open or closed with associated energy 0 if closed and ##\epsilon## if open. a) Suppose the N links are independent, compute the partition function of the system and the average number of open links b)Now assume that...
  3. C

    Laplace Transforms: Transfer Functions and Impulse

    Homework Statement I uploaded the question as a picture and attached it. Homework Equations Unit step function - u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} Impulse function - δ(t) = \displaystyle\lim_{Δ\rightarrow 0} δ_Δ (t) Multiplication Property...
  4. I

    MATLAB Simple vector functions in matlab

    I have an assignment for MATLAB where I am required to create two vectors x and y and then to find the sum in three ways. first, create an extra variable z and find the sum second, use the dot function third, multiply x with the transpose of y here's my code x=linspace(0,1,5)...
  5. C

    Laplace Transforms: Transfer Functions, and IVT/FVT Problems

    Homework Statement I uploaded the problem statements as a picture as well. I have completed these and was wondering if someone could check my work, and let me know if it is correct. Problem 1.3: Find the expression for the transfer function of this linear time-invariant causal system with...
  6. Prof. 27

    Function Composition of Multivariate Functions

    Homework Statement This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...
  7. R

    Position Vector in Wave Functions

    Hello, I need to create a 2-D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the superposition of the symmetric and anti-symmetric wave functions, $$\Psi_s(\textbf{r}) =...
  8. naima

    Test functions in wightman axioms

    According to wikipedia AQFT needs test functions so that the fields are distributions smeared on these functions. I'd want to know what are these test functions. I read in Haag's book that they are fast decreasing functions defined on space time. They belong to the set S of Schwartz functions...
  9. J

    How to find the equation of this tangent?

    Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...
  10. T

    Can fourier sine series approximate even functions?

    I am learning Fourier series and have come across the sine, cosine, and imaginary exponential expressions. To my knowledge, these individual terms form a basis since they are all orthogonal to each other. I am just wondering: can a Fourier sine series be used to model a purely even function...
  11. J

    Book on gamma functions with applications in Quantum Mech.

    I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions. Does anybody know a book(or any other source) that I can learn about and practice gamma functions integration...
  12. C

    Laplace Transforms Involving: Unit-Step, and Ramp Functions

    Homework Statement Here is an imgur link to my assignment: http://imgur.com/N0l2Buk I also uploaded it as a picture and attached it to this post. Homework Equations u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} The Attempt at a Solution Question 1.1 -...
  13. E

    Grade 12 Math (inverse of functions)

    f(x)=[((x-1)/(x+1))+((x-1)/(x+1))]1/2 What is F-1(x) No matter what I try I am unable to isolate for y after switching x's with y's. Any ideas?
  14. 24forChromium

    Use Excel to find the similarity between functions.

    I am writing a paper and I came up with a function that uses a hypothetical relationship to predict the value of one variable at different points in time, I graphed it, and then graphed the actual readings from an experiment. How can quantitatively describe how close the two trends are? In other...
  15. N

    Orthogonal properties of confluent hypergeometric functions

    Hi Can anyone point to me a reference where orthogonal properties of confluent hypergeometric functions are discussed? Navaneeth
  16. M

    Integration: inverse trigonometric functions

    Homework Statement ∫(t/√(1-t4))dt Homework Equations ∫(du/√(a2 - u2)) = arcsin (u/a) + C ∫(du/(a2 + u2) = (1/a) arctan (u/a) ∫(du/(u√(u2 - a2))) = (1/a) arcsec (|u|/a) The Attempt at a Solution Edit: I meant to write u where t2 is[/B]
  17. Ackbach

    MHB What are Bessel Functions and how can they help solve differential equations?

    This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download. Bessel Functions $$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$...
  18. W

    Solving Functions f and g: Range and Relationship with Homework Statement

    Homework Statement The functions f and g are given by https://scontent-kul1-1.xx.fbcdn.net/hphotos-xtf1/v/t35.0-12/11993948_10204837479208369_1887096410_o.jpg?oh=b1653a61128c571af8137b1fd00ccb01&oe=55F47BC4 a) without using differentiation, find the range of f b)show that f(x)^2+g(x)^2=1.Hence...
  19. Math Amateur

    MHB Ideal of functions disappearing at (a_1, a_2, .... .... , a_n)

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help me to...
  20. J

    Distributions & test functions in specific applications

    I have a question inspired by a recent thread that I did not want to hijack (https://www.physicsforums.com/threads/distributions-on-non-test-functions.831144/) I realize that weaker requirements on the space of test functions results in a more restricted set of distributions. For example, if...
  21. O

    MHB Increasing, non-increasing, decreasing and non-decreasing functions

    Please can you give definitions of increasing, non-increasing, decreasing and non-decreasing functions ? I found something but there is a lot of differents between these definitions...Can you give these definitions ? Thank you so much, Best wishes :)
  22. pellman

    Distributions on non-test functions

    The definitions of distributions that I have seen (for instance https://en.wikipedia.org/wiki/Distribution_(mathematics)#Distributions ) define a distribution as a map whose domain is a set of test functions. A defining quality of test functions is that they have compact support, which for most...
  23. M

    MHB Can Absolute Values of Quadratic Functions Determine Their Discriminants?

    Let f(x) and g(x) be quadratic functions such as the inequality \left| f(x) \right| \ge \left| g(x) \right| is hold for all real x . Prove that \left| \Delta_f \right| \ge \left| \Delta_g \right|. For quadratic function f(x)=ax^2+bx+c , then \Delta=b^2-4ac. I have no idea how I could...
  24. Y

    Modeling a Control System using Transfer Functions

    Homework Statement [/B] Homework Equations Listed under 2.1 in the image above. This is the only relevant equation that I'm aware of, but I'm almost sure that there is something else I need to know before I can solve the problem. The Attempt at a Solution I tried solving for the...
  25. DeldotB

    Show a functions inverse is injective iff f is surjective

    Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.
  26. BruceW~

    Help with Listing Functions w/ Same Range & Diff Domains

    Homework Statement question: State two different functions that have same range but different domain. Then tell me what is the range of those two functions. The attempt at a solution Y = x/2 Y= x/2 +1 I don't know if that is correct or not. Any suggestion will help.
  27. jk22

    Measurement problem and computer-like functions

    Suppose we define the measurement of an observable A by v(A) v being an 'algorithm giving out one of the eigenvalues each time it is called' (we accept the axiom of choice) In this context we have in particular v(A)≠v(A) since when we call the left hand side and then the right handside the...
  28. DaniV

    The Trigonometery functions in other presentation

    Can I present the trigonometry functions (such as SinX, CosX, TanX, CotX) by using only- Log, Lan, multiplication, division, addition, subtraction, exponantion, nth root operations?
  29. P

    Proving Inverse Functions: Multiplicative Relationships

    Is there a way to formally prove that if ##f## and ##g## are multiplicative inverses of each other, then ##f^{-1} (x) = g^{-1} (\frac{1}{x})##?
  30. Destroxia

    What are the common functions used to solve limits in single variable calculus?

    Homework Statement Can you create a list of which functions increase towards infinity the fastest for limit solving? Homework EquationsThe Attempt at a Solution I'm trying to make a list from least speed, to fastest speed, in approaching infinity. As in, if you have a limit, and it has...
  31. Mr Davis 97

    Order of transformation of functions?

    I am confused about the order in which we apply transformations to a input of a parent function to get the corresponding input of the new function. Say for example, we have the function ##y = \sin(-2x + 1)=\sin(-2(x-\frac{1}{2}))##. Intuitively, it would seem as though we would transform a point...
  32. Math Amateur

    MHB Rational Functions - Polynomials Over a Field - Rotman Proposition 3.70

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...
  33. L

    Integral involving exponential functions

    Please help give references on solving the following integral: \int\frac{1}{c_{1}e^{ax}+c_{2}e^{bx}}dx where a\neq b Thanks a lot in advance.
  34. pellman

    Are all smooth functions square-integrable?

    Came across this in a discussion of essential self-adjointedness: Let P be the densely defined operator with Dom(P) = C^{\infty}_c (\mathbb{R}) \subset L^2 ( \mathbb{R} ) and given by Pf = -i df/dx. Then P is essentially self-adjoint. It is the C^{\infty}_c (\mathbb{R}) \subset L^2 (...
  35. W

    Entire Functions and Lacunary Values.

    #Hi All, Let ## f: \mathbb C \rightarrow \mathbb C ## be entire, i.e., analytic in the whole Complex plane. By one of Picard's theorems, ##f ## must be onto , except possibly for one value, called the lacunary value. Question: say ##0## is the lacunary value of ##f ##. Must ## f ## be of the...
  36. Titan97

    Checking if f(x)=g(x)+h(x) is onto

    This is picture taken from my textbook. I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...
  37. Titan97

    Prove that [a/b]+[2a/b]+....+[(b-1)a/b]=(a-1)(b-1)/2

    Homework Statement Prove that $$\sum_{r=1}^{b-1}[\frac{ra}{b}]=\frac{(a-1)(b-1)}{2}$$ where [.] denotes greatest integer function and a & b have no common factors. Homework Equations ##n\le [n]<n+1## <x> denotes fractional part of x. 3. The Attempt at a Solution I first added and subtracted...
  38. T

    Are set theory functions sets too?

    I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B. But set theory has functions too, could they be sets too...
  39. W

    Periodic Functions: Is Irrationality the Cause of Non-Periodicity?

    Hey. Assume you have a signal ##f## with period ##T_f## and a signal ##g## with period ##T_g##. Then the signal ##h= f+g## is periodic iff ##T_f/T_g \in \mathbb{Q}##. So if ##T_f/T_g## is an irrational number, the signal ##h## will not be periodic. Why is this actually the case?
  40. Mr Davis 97

    Is computation synonymous with functions in computer science?

    So I've delved into programming, and gotten experienced with the fundamentals. However, the more I learn, the more I question the central object of comp. science, computation, and its foundation. According to Wikipedia, "Computation is any type of calculation that follows a well-defined model...
  41. ognik

    MHB Reverse direction for complex functions

    Hi An exercise asks to show $ \int_{a}^{b}f(z) \,dz = -\int_{b}^{a}f(z) \,dz $ I can remember this for real functions, something like $ G(x) = \int_{a}^{b}f(x) \,dx = G(b) - G(a), \therefore \int_{b}^{a}f(x) \,dx = G(a) - G(b) = -\int_{a}^{b}f(x) \,dx $ I have seen 2 approaches, either...
  42. Amrator

    Rate of Change Using Inverse Trig Functions

    Homework Statement A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...
  43. W

    Tight Binding Wave Functions

    Dear all, Could somebody please, indicate me some tutorial, in order to generate a 3D grid to plot the wave function using the Hamiltonian eigenvalues and the slater type orbitals ? Thanks in advance, Wellery
  44. C

    Writing correct mathematics -- functions within functions....

    Hi I'm a bit confused about some mathematical notation If i write f(x)=(2x^2 + 10)^4 And i define u= 2x^2 +10 u^4 = f(x) Would it then be correct to write f(u)= u^4 Or would i get f(u)= 2(u)^2 +10 = (2(2x^2 +10)+10)^2 Should i define u^4 = f(x) first? Would it then be correct...
  45. F

    Linear Algebra vector functions LI or LD

    Homework Statement Determine whether or not the vector functions are linearly dependent? u=(2t-1,-t) , v= (-t+1,2t) and they are written as columns matrixes. Homework Equations Wronskian, but I don't think I should use it because I need to take derivatives so it doesn't seem like it would...
  46. ognik

    MHB Do Cauchy-Riemann Conditions Guarantee Analyticity?

    Hi - just started complex analysis for the 1st time. I have been a little confused as to the chicken and egg-ness of Cauchy-Riemann conditions... 1) Wiki says: "Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann...
  47. C

    Mixing units with functions or derivatives?

    Hi, How do you correctly use units when writing derivatives and functions in math? Example A car goes 17miles per gallon, so a function m with the equation m(g)=17g describes the distance it can go with g gallons. And the derivative dm/dg = 17 miles/gallon. Question: could you write the...
  48. E

    Re-scaling Functions under the Same Axes

    Consider two functions ##f\left(x, y\right)## and ##g\left(px, qy\right)##, where ##p## and ##q## are known. How can I plot the two functions on the same graph (i.e. the same axes)? The function ##f\left(x, y\right)## will have axes with values ##x## and ##y##, while the other will have axes...
  49. D

    Sum of Related Periodic Functions

    I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question: Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c...
  50. Math Amateur

    MHB Polynomials and Polynomial Functions in I_m = Z/mZ

    I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3. Polynomials I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} The relevant section...
Back
Top