Functions Definition and 1000 Threads

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. M

    MHB What Is the Domain and Range of y=cos(3(x - 45°)) +2?

    State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.
  2. M

    MHB What are the first two positive x-intercepts for the given sinusoidal function?

    Find the first two positive x-intercepts for y= -2cos(3(x-25°)) +1 (Can someone help me for this)
  3. M

    MHB Sinusoidal Functions (I for this)

    Sinusoidal Functions... Can someone help me with this. Describe the transformations that are applied to y= -4cos[2(x-30°)] +5 (State any shifts, stretches, compressions, or reflections).
  4. Arman777

    Python Functions, naming conventions in Python

    Sometimes, when I code something, I am naming the local variables in the function same as the global variable. Such as, my_var = 13 def iseven(my_var): if my_var % 2 == 0: return True return False print(iseven(my_var)) As you can see my_var is defined globally but also used...
  5. L

    B Commutators of functions of operators

    I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute? Also I have a question is it possible that ##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
  6. Eclair_de_XII

    Integrating vector-valued functions along curves

    The following parametrizations assume a counter-clockwise orientation for the unit square; the bounds are ##0\leq t\leq 1##. Hypotenuse ##(C_1)## %%% ##r(t)=(1-t,1-t)## ##dr=(-1,-1)\,dt## ##f(r(t))=f(1-t,1-t)=(a(1-t)^2,b(1-t)^2)## ##f\cdot dr=-(a+b)(1-t^2)\,dt## \begin{align} \int_{C_1} f\cdot...
  7. C

    I Longitudinal DIS Structure functions

    DIS observables can be expressed in terms of structure functions F1, F2 and FL. There exists the relation ##F_L = F_2 - 2xF_1##. We can write $$ F_L = \sum_a x \int_x^1 \frac{dy}{y} C_{a,L}(y,Q) f_a (\frac{x}{y},Q) $$ and similarly for ##F_1## and ##F_2##: $$ F_1 = \sum_a x \int_x^1...
  8. LCSphysicist

    Can Laplace Transforms be Applied to Finite Intervals?

    "Consider a string of length L that is connected at both ends to supports and is subjected to a load (external force per unit length) of f(x). Find the displcament u" https://i.stack.imgur.com/yVIDG.png We need to solve this: $$Tu_{xx} = f(x)$$ subject to $$u(0)=u(L)=0$$ But i don't understand...
  9. E

    MHB How to Solve a Composition of Functions g(f(x))?

    g(f(x))=\( g(f(x))=2/(1/x-4)+4 \)
  10. S

    MHB Fourier Series involving Hyperbolic Functions

    Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
  11. N

    MHB Exercise about the concept of functions

    Hi everyone! =) . I'm having some issues with this exercises, It's about functions. I remember the basic geometrics formulas and how to get the area and perimeter of a square or a circle but I don't get it. I need an explanation. 1. Express the area A of a square as a function of (a) the length...
  12. C

    I Cardinality of decreasing functions from N to N

    Problem: Find the cardinality of the set ## A = \{f \in \Bbb N \to \Bbb N. \forall n\leq m .f(n) \geq f (m) \} ##. I know that ## A \subseteq P(\Bbb N \times \Bbb N) ## implies ## |A| \leq |P(\Bbb N \times \Bbb N)| = | P(\Bbb N) | = \aleph ##. So I have a feeling that ## \aleph \leq |A| ##...
  13. S

    A Multinomial functions of matrices

    What branch of mathematics studies multinomial functions of matrices? ( i.e matrix valued functions of square matrices such as ##f(A,B,C) = ABC + BAC + 2A^2 + 3C##)
  14. 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...
  15. 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...
  16. 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}...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. D

    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...
  24. 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!
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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.
  30. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  31. 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...
  32. 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) =...
  33. 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?
  34. 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...
  35. 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...
  36. 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?
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. G

    B What Are Functions Called That Are Linearly Dependent With Their Derivatives?

    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?
  42. 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 ...
  43. 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...
  44. 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...
  45. 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...
  46. 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}
  47. 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 = \{...
  48. 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...
  49. 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...
  50. 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...
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