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. N

    MHB Which procedure takes the minimum time to solve modulus functions?

    1) -|2x-3|+|5-x|+|x-10|=|3-x| 2) |2x-3|-|5-x|-|x-10|-|3-x|=28 3) -|2x-3|+|5-x|+|x-10|≥|3-x| How can we solve these problems? The method I know is to plug in the critical values to see which modulus becomes positive and which one becomes negative. Then find out the values of x for which the...
  2. D

    A Calculating nonequilibrium Green's Functions

    Hey :) Firstly I want to thank everyone who takes their time to read through this post and who tries to help me. So the issue is the following: I wrote a python code that creates a Lindbladian, and I wanted to try to calculate the Greens function using the Lehmann representation. For the...
  3. C

    Finding inverses of two functions in Lambda Notation

    I found the following functions ( In lambda notation ) to be injective, and now I am trying to find the inverse functions for them ( the inverse for the Image of ## f ## ) but I am stuck and I need help: 1. ## f = \lambda n \in \mathbb{N}. (-1)^n + n^2 ## 2. ## f = \lambda g \in \mathbb{R}...
  4. Eclair_de_XII

    B Are continuous functions on sequentially compact sets u-continuous?

    Suppose ##f## is not uniformly-continuous. Then there is ##\epsilon>0## such that for any ##\delta>0##, there is ##x,y\in K## such that if ##|x-y|<\delta##, ##|f(x)-f(y)|\geq \epsilon##. Choose ##\delta=1##. Then there is a pair of real numbers which we will denote as ##x_1,y_1## such that if...
  5. H

    Determining the growth of two functions using Big-Oh definition

    My attempt involved using the big-Oh notation, I think this should work but I am not sure how to go about it. The two functions are g(n) = 6^n/n^5 and h(n) = (ln n)^84. I thought that I could use the inequality 6^n < ln(n)^84 and 6^n/|n^5| = |g(n)| < 6^n and put those inequalities together...
  6. garthenar

    Engineering W0 = 1/RC? Transfer Functions and Bode Plots

    Here is the example and solution in full. I have circled where I'm at and highlighted the part that's tripping me up. I managed to get... and getting everything in terms of the angular frequency seems to be critical for getting the plots for the Frequency Response. I checked my notes on RC...
  7. D

    Tetrahedron Simplex Shape Functions in FEA

    Hi, 2 part question trying to get tetrahedron Finite Element shape functions working: 1) How do I properly setup the shape coefficient matrix and 2) How do I build the coefficient quantities in the shape functions properly? ANY tips or corrections may unblock me and would be of much value...
  8. K

    I Transition Functions and Lie Groups

    I understand that on Riemannian manifolds, the transition functions that glue charts together are coordinate transformations (Jacobian matrices). However, I am not quite sure how transition functions work in the context of Lie groups and Fiber bundles. Do we consider the manifolds to be flat...
  9. MathematicalPhysicist

    Mathematica Series expansion from the red book on special functions by Richard Ask

    I want to check my calculations via mathematica. In the book I am reading there's this expansion: $$\frac{(1+\frac{1}{j})^x}{1+x/j}=1+\frac{x(x-1)}{2j^2}+\mathcal{O}(1/j^3)$$ though I get instead of the term ##\frac{x(x-1)}{2j^2}## in the rhs the term: ##-\frac{x(x+1)}{2j^2}##. So I want to...
  10. docnet

    Find the set of all functions that satisfy the inequality

    Problem: Find the set of all harmonic functions ##u(x,y,z)## that satisfy the following inequality in all of ##R^3## $$|u(x,y,z)|\leq A+A(x^2+y^2+z^2)$$ where ##A## is a nonzero constant. Work: I removed the absolute value bars by re-writing the expression $$-C-C(x^2+y^2+z^2)\leq u\leq...
  11. B

    Show that f such that f(x+cy)=f(x)+cf(y) is continuous

    We need to show that ##\lim_{x \rightarrow a}f(x)=f(a), \forall a \in \mathbb{R}## . At first, I tried to show that f is continuous at 0 and from there I would show for all a∈R. But now, I think this may not even be true. I only got that f(0)=0. I'm very confused, I appreciate any help!
  12. Strand9202

    Concavity and Tangent Functions

    Here is the problem (8b). I was asked to write out why the circled part was true. I know that since the function is concave down then f"(x)<0. That is a fact. What I am having trouble with is why they can say the next part. What I thought was L(x) is the tangent line and all tangent lines...
  13. JD_PM

    Loop Feynman diagram contributions to correlation functions

    My understanding of the n-correlation function is \begin{equation*} \langle \phi(x_1) \phi(x_2) ... \phi(x_n)\rangle = i \Delta_F (x_1-x_2-...-x_n) \end{equation*} Where ##\Delta_F## is known as the Feynman propagator (in Mathematics is better known as Green's function). Let us analyze...
  14. J

    Can the limit of a quotient of trig functions approach a specific value?

    Hello. Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist. But can a quotient of the two acutally approach a certain value? lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...
  15. M

    A Example of Ritz method with Bessel functions for trial function

    Hi PF! Do you know of any examples of the Ritz method which use Bessel functions as trial functions? I’ve seen examples with polynomials, Legendre polynomials, Fourier modes. However, all of these are orthogonal with weight 1. Bessel functions are different in this way. Any advice on an...
  16. S

    MHB Determine the area of a region between two curves defined by algebraic functions

    R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.
  17. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  18. L

    MHB Understanding O(h): Examining Functions with h as an Input

    Let \mid h \mid < 1. Which of the following functions are O(h)? Explain. -4h h+h^2 \mid h \mid ^{0.5} h + cos (h) Based on my notes, f(h) = O(h) only if \mid f \mid ≤ C \mid h \mid , where C is a constant independent of h. I can only solve for the first function -4h, as I can...
  19. M

    Probability Density Functions: Transformation of Variables

    Hi, I have a question about probability transformations when the transformation function is a many-to-one function over the defined domain. Question: How do we transform the variables when the transformation function is not a one-to-one function over the domain defined? If we have ## p(x) =...
  20. John Greger

    I Units of trigonometric functions?

    What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?
  21. S

    Mathematica Plotting 2 functions & economizing on calc of intermediate result

    I am trying to plot two functions a(t) and b(t) that both use a common intermediate result K. In my actual code K would be a slow-ish calculation. To reuse the K across a and b, I am putting them into a single module that provides an array {a, b} to the Plot[] function. (BTW, the Evaluate[] is...
  22. S

    I Extremums of functions: x to the power of x to the power of a

    I plotted the x(red dots) coordinates and y coordinates( black dots) of extremums of functions x^x^a (the x coordinate of the dots is a and y coordinate of the dots is x or y coordinate of the extremum). Is there a function, on which are located all the black dots or all the red dots? P.S. The...
  23. S

    Mathematica Ratio of functions -- automatically apply l'Hospital rule when needed

    I want to plot the ratio f1(x) / f2(x), where they have some common zeros. Does Mathematica have a feature that will do this, switching automatically to f1'(x) / f2'(x) when appropriate, avoiding F.P. errors and optimizing numerical precision? If not, is there a good way to implement this?
  24. yucheng

    B Significant figures for special functions (square roots)

    I am using square roots, however, I am confused over how many significant figures (s.f.) to keep. Suppose I have ##\sqrt{3.0}##, which has 2 s.f. From three different sources, I'll put a summary in brackets: https://www.kpu.ca/sites/default/files/downloads/signfig.pdf (if 2 s.f. in the data...
  25. Mayhem

    I Showing that a set of differentiable functions is a subspace of R

    Problem: Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}## This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...
  26. F

    Vector space of functions from finite set to real numbers

    Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...
  27. evinda

    MHB Master Theorem-Relation of the two functions

    Hello! (Wave) I want to solve the recurrence relation $$T(n)=4T{\left( \frac{n}{3} \right)}+n \log{n}.$$ I thought to use the Master Theorem. We have $a=4, b=3, f(n)=n \log{n}$. $\log_b{a}=\log_3{4}$ $n^{\log_b{a}}=n^{\log_3{4}}$ How can we find a relation between $n^{\log_{3}{4}}$ and...
  28. G

    B Name for sine-like functions

    Is there a name for functions that are linearly dependent with its derivatives? i.e. a function ##f(x)## such that, for some value of ##n## it fulfills $$f^{(n+1)} = \sum_{k=0}^{n} \alpha_k f^{(k)}$$ are linearly dependent?
  29. F

    I Proving linear independence of two functions in a vector space

    Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
  30. L

    MHB Compose Functions: True/False?

    Which of the following are true? Select all options. Assume that f:A→B and g:B→C. If f and g are injective, then so is g∘f If f and g are surjective, then so is g∘f If f and g are bijective, then so is g∘f If g∘f is bijective, then so are both f and g If g∘f is...
  31. M

    Exploring Variable Slopes in Advanced Functions

    So I attempted this problem and to satisfy the first condition (for t in the range of [1, 5]), I drew the straight line that has a slope of 5 (i.e. f(x)=5x). I just don't understand how I can have the same function with a different slope (average rate of change) for the interval [1,10] or for [2...
  32. S

    I Eigenfunctions and wave functions

    I saw this statement from the textbook "Quantum physics of atoms, molecules, solids, nuclei, and particles" second edition pg 166. According to the text, is the author saying the solution to the TISE is the eigenfunction and when you multiply the time dependent part, you get the wave function? I...
  33. K

    MHB Matrix Transformation - mappings of functions

    I need to find the matrix transformation of y = \frac{1}{x} onto y = \frac{-1}{3x-1}-2 I think its \begin{bmatrix} x'\\ y' \end{bmatrix} =\begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} -1\\ -2 \end{bmatrix}
  34. E

    B Why does f = g in implicit equations?

    I saw something in my notes that I didn't understand... we have ##y=f(x)##, and consider an implicit equation of the form ##g(y) = f(x)##. They then say that ##f=g##. Why is that true? I would have thought$$f = \{ (a,f(a)) : a\in \mathbb{R} \} \subseteq \mathbb{R}^2$$whilst ##g## is just$$g = \{...
  35. joneall

    I Where do wave functions come from?

    In classical mechanics, we have either Newton’s laws or a Lagrangian in terms of coordinates and their derivatives (or momenta) and we can solve them for the behavior of the system in terms of these variables, which are what we observe (measure). In QM, we quantize classical mechanics by making...
  36. M

    Inverse trigonometric functions

    Create one equation of a reciprocal trigonometric function that has the following: Domain: ##x\neq \frac{5\pi}{6}+\frac{\pi}{3}n## Range: ##y\le1## or ##y\ge9## I think the solution has to be in the form of ##y=4sec( )+5## OR ##y=4csc( )+5##, but I am not sure on what to include...
  37. T

    I Limits of functions and sequences

    Hello there.Is there any function or sequence that has no limits at any point? I am not necessarily talking about functions on euclidean spaces, they could be on topological spaces in general.Also, we have homeomorphism that is about I think mostly continuity, diffeomorphism about...
  38. AN630078

    Composite and Inverse Functions Problem

    1. a. fg(x)=2(1/2(x-1))+1 fg(x)=2(x/2-1/2)+1 fg(x)=x-1+1 fg(x)=x gf(x)=1/2((2x+1)-1) gf(x)=1/2(2x+1-1) gf(x)=x+1/2-1/2 gf(x)=x The functions functions f(x) and g(x) are inverses of each other. This can be demonstarted by f(x)=2x+1 y=2x+1 x=2y+1 x-1=2y (x-1)/2=y Thus, y=1/2(x-1) = g(x) And...
  39. M

    Advanced functions (precalculus)

    This is my attempt so far: ##0.05=\frac{30t}{200000+t}## then I solved for t. And I got 333.88 min. I feel like this is way too simple of a solution and I didn't use all of what's given in the problem. For part 2 of the problem it asks, what happens to the concentration over time. I tried to...
  40. F

    I Asymptotes of Rational Functions....

    Hello, I know that functions can have or not asymptotes. Polynomials have none. In the case of a rational functions, if the numerator degree > denominator degree by one unit, the rational function will have a) one slant asymptote and b) NO horizontal asymptotes, c) possibly several vertical...
  41. banananaz

    MHB How do I find the Euclidean Coordinate Functions of a parametrized curve?

    I've been given a curve α parametrized by t : α (t) = (cos(t), t^2, 0) How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here. (Sorry about the formatting)
  42. srfriggen

    Comparing the domains of composite functions

    Hello, I have attached my question and the work. I believe the answer is correct. Looking for verification. Thank you!
  43. Sabertooth

    I Elliptic Function Rotation Problem

    Hi all:) In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion. Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane. When I plot a point rotating...
  44. S

    I Convergence of sequences of functions with differing domains?

    Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...
  45. Sabertooth

    Expressing Elliptic Orbitals As Speed Functions.

    Hi everyone:) I have spend a couple of days trying to teach myself the math of orbital mechanics and have been able to generate a model of the orbital path of Haley's Comet, incorporating realistic distances and periods using Kepler's second law & ellipsoid functions. This is a GIF of the motion...
  46. M

    I Derivatives of Standard Functions

    I can't help but feeling these days that I don't actually understand where most of the maths I use comes from. Unfortunately, I can't remember whether this is due to the fact that I didn't take my studies seriously until the end of undergrad, or rather that these things were never actually...
  47. Riccardo Marinelli

    Initial condition of Wave functions with Yukawa Potential

    Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...
  48. M

    MHB Sequence of functions : pointwise & uniform convergence

    Hey! 😊 Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$. For an integer $m>a$ we have that \begin{equation*}0 \leq...
  49. madafo3435

    I Differential analysis: convex functions

    I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they...
  50. xyz_1965

    MHB How do trigonometric functions and their inverses relate to each other?

    Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)? Why does arcsin (sin x) = x? Can it be that trig functions and their inverse undo each other?
Back
Top