Functions Definition and 58 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. P

    Biology Which organs/parts of the body are only functional on glucose?

    Hi everyone! Do you have an idea which organs/parts of the body are ONLY functional on glucose? I would say the brain, pancreas, liver and kidney, but I have to take into account only those organs that are ONLY functional on glucose
  2. MevsEinstein

    B Recurrence relation in a recurrence relation?

    There's a famous functional equation that was asked in the 2019 IMO. It looks like this: find all f: Z -> Z where f(2a)+2f(b)=f(f(a+b)). I thought of solving it using a recurrence relation where a_n=f(nx). But when I substituted values in the functional equation (after setting a and b equal...
  3. aspiringastronomer

    Struggling in my freshman year of Physics at university

    If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it? Hello...
  4. pairofstrings

    B Arithmetic progression, Geometric progression and Harmonic progression

    How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!
  5. 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!
  6. 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...
  7. Physics lover

    No. of solutions of an equation involving a defined function

    Here is a pic of question My attempt-: I defined functions f(x-1) and f(x+1) using f(x).After defining them,I substituted their values in the equation f(x-1)+f(x+1)=sinA. For different ranges of x,I got different equations. For 1<x<2,I got 1-x=sinA. But now I am confused.For each different...
  8. brotherbobby

    Proving that the two given functions are linearly independent

    Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...
  9. brotherbobby

    I Proving functions are linearly dependent

    We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##. However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I...
  10. B

    I Finding CDF given boundary conditions (simple stats and calc)

    I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
  11. ubergewehr273

    Question about a function of sets

    Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
  12. T

    Two Limit exercises of functions of two variables.

    (I) Find the limit (x,y)->(0,0) of F, then prove it by definition. (II) Find the limit and prove it by definition of: as (x,y) approach (C,0), C different from zero. I have previously asked it on Quora, but it doesn't appear to have answers any...
  13. SemM

    I How to find admissible functions for a domain?

    Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...
  14. C

    I Is there a geometric interpretation of orthogonal functions?

    Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
  15. C

    Comp Sci Book Database Implementation C++

    I began by creating 2 classes. A book class and a course class that contains any necessary info about the book and course respectively class bookClass{ private: string theISBN; string thebookName; string thebookAuthor; double thebookCost; int...
  16. Oats

    I Must functions really have interval domains for derivatives?

    Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
    • Oats
    • Thread
    • analysis derivative domain functions interval
    • Replies: 2
    • Forum: Calculus
  17. M

    Finding dy/dx for a circle

    Homework Statement Hello I have this circle with the equation : [/B] (x-a)^2+(y-b)^2=r^2 I want to find dy/dx for it 2. Homework Equations (x-a)^2+(y-b)^2=r^2 The Attempt at a Solution I am looking on the internet and it appears that I should use what is called "Implicit differentiation"...
  18. D

    Problem regarding periodic current functions

    Homework Statement Three periodic currents have the same ##f=100 Hz##. The amplitude of the second current is ##4 A##. and is equal to half of the amplitude of the third current. Effective value of the third current is 5 times that of the first current. At time ##t_1=2ms## third current...
  19. Bunny-chan

    Book demonstration about trigonometric relations

    Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution
  20. K

    B Average angle made by a curve with the ##x-axis##

    The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$ I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...
    • Kumar8434
    • Thread
    • angle derivatives functions integration trigonometry
    • Replies: 4
    • Forum: General Math
  21. K

    I Expression of ##sin(A*B)##

    Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
    • Kumar8434
    • Thread
    • functions integral taylor series trigonometry
    • Replies: 19
    • Forum: General Math
  22. K

    B Expressions of ##log(a+b), tan^{-1}(a+b),sin^{-1}(a+b)##,etc

    Hi, I got these: $$log(a+b)\approx \frac{b*logb-a*loga}{b-a} + log2 -1$$ $$tan^{-1}(a+b)\approx \frac{b*tan^{-1}2b-a*tan^{-1}2a+\frac{1}{4}*ln\frac{1+4a^2}{1+4b^2}}{b-a}$$ $$sin^{-1}(a+b)\approx \frac{b*sin^{-1}2b-a*sin^{-1}2a+\frac{1}{2}*(\sqrt{1-4b^2}-\sqrt{1-4a^2}}{b-a}$$ And, similarly for...
  23. K

    B Find ##f(x)## such that f(f(x))=##log_ax##

    I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...
  24. E

    MATLAB Does MatLab have this kind of function?

    I have a set of variables that are always inputs for several functions that I made. Does MatLab have a kind of function that stores these variables into a single matrix (or similar) so that I just need to call this matrix for each function rather than calling them one-by-one as inputs into the...
  25. Q

    I Analysis of a general function with a specific argument

    Hello everybody, I'm currently helping a friend on an assignment of his, but we are both stumbled on this exercise, I'm posting it here We define a function ##f## which goes from ##\mathbb{R}## to ##\mathbb{R}## such that its argument maps as $$ x \mapsto...
  26. Stoney Pete

    I Can an ordered pair have identical elements?

    Hi guys, Here is a wacky question for you: Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written...
  27. S

    I Differentiability of multivariable functions

    What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
    • Sho Kano
    • Thread
    • continuity differentiability functions limit multivariable
    • Replies: 9
    • Forum: Calculus
  28. Z

    B Help with understanding Nature of Roots for Quadratic and Cu

    Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to...
  29. M

    Inverse function help

    Homework Statement Let f(x) = 1−3x−2x^2 , x ∈ [−2, −1]. Use the Horizontal Line Test to show that f is 1–1 (on its given domain), and find the range R of f. Then find an expression for the inverse function f −1 : R → [−2, −1]. The Attempt at a Solution I have already done the horizontal line...
  30. M

    Trig function help

    Homework Statement Let C=cosx. Write sec(2x)csc(x)sin(2x) as a function of C. The Attempt at a Solution Am I on the right track 1/cos(2x) * 1/sin(x) * 2sin(x)cos(x) 1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x) What would i do from here?
  31. M

    Domain and range help

    Homework Statement Suppose f is a function with domain [-2,10] and range [5,10]. Find the domain and range of the following functions. (a) f(2x+4) (b) 2f(x)+4 The Attempt at a Solution [/B] Would I just substitute the in the domain and range values to find the answer?
  32. I

    MATLAB Transforming part of matlab code to Fortran90

    Here are my Fortran codes: program test implicitnone integer*4 nxProjPad, cf, numViews, cc, index, indRad, iv, i, INDEX1, d, n real*4 v4, v5, RSS, S1, F1, gMDL real*4, dimension(:), allocatable :: array, sum, cumsum, transpose, log, SS1, SSs nxProjPad=185 numViews=180...
  33. MrDickinson

    I Can someone me simplify this expression....

    lim_(h->0^-) (e^(x+h)/((x+h)^2-1)-e^(x+h)/(x^2-1))/h = -(2 e^x x)/(x^2-1)^2 I know how to differentiate the expression using the quotient rule; however, I want to use the limit definition of a derivative to practice it more.This desire to practice led me into a trap! Now I just can't simplify...
    • MrDickinson
    • Thread
    • calculus derivation e^x functions limit
    • Replies: 8
    • Forum: Calculus
  34. C

    No of ordered pairs satisfying this equation

    Homework Statement We are required to find the no. of ordered pairs ##(x,y)## satisfying the equation ##13+12[tan^{-1}x]=24[ln x]+8[e^x]+6[cos^{-1}y]##. (##[.]## is the greatest integer function, e.g. ##[2.3]=2##, ##[5.6]=5##, ##[-2.5]=-3## etc) Homework Equations The Attempt at a Solution...
  35. C

    Maxima of discrete functions involving nPr, nCr, etc?

    Homework Statement So I want to prove that the expression 20Cr×0.1r 0.9(20-r) reaches maximum value for r=(0.1)×20=2 Homework Equations The Attempt at a Solution I can prove it by trial and error but can't differentiate the expression because nCr isn't continuous.
  36. KevinFr

    B Finding intervals of a 3 degree function?

    The question says find apex, low point and the monotonic properties of the functions. a) b) c)... To find intervals, I use the abc-formula. Example: f(x) = 3x^3 - 3x d/dx * f(x) = 3 * 3x^2 - 3, here a=3*3, b= -3 and c=0 (because there is none) x1 = ( -b + sqrt(b^2 + 4*ac) ) / 2a x2 = ( -b -...
    • KevinFr
    • Thread
    • calculus functions math
    • Replies: 4
    • Forum: Calculus
  37. V

    Prove the integral is in the range of f

    Homework Statement If f: [0,1] \rightarrow \mathbb{R} is continuous, show that (n+1) \int_0^1 x^n f(x) \mathrm{d}x is in the range of f Homework Equations (n+1) \int_0^1 x^n f(x) \mathrm{d}x=\int_0^1 (x^{n+1})' f(x) \mathrm{d}x The Attempt at a Solution I tried integration by parts, but that...
  38. Titan97

    Number of functions such that f(i) not equal to i

    Homework Statement ##A=\{1,2,3,4,5\}##, ##B=\{0,1,2,3,4,5\}##. Find the number of one-one functions ##f:A\rightarrow B## such that ##f(i)\neq i## and ##f(1)\neq 0\text{ or } 1##. Homework Equations Number of derangements of n things =...
  39. N

    Plotting functions

    if we draw a line parallel to the x- axis and passes through a point in the image and the graph intersects at one point is this a one to one function ?
  40. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
  41. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
  42. W

    Finding the PDF and CDF of a given function Z = X/Y

    Homework Statement Given a Uniform Distribution (0,1) and Z = X/Y Find F(z) and f(z) Homework Equations The Attempt at a Solution So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it. R(z) = {0,∞} because as y is very small...
  43. lordianed

    Prove that no such functions exist

    Homework Statement Prove that there do not exist functions ##f## and ##g## with the following property: $$(\forall x)(\forall y)(f(x+y) = g(x) - y)$$ Homework Equations NA The Attempt at a Solution Here is some information I have found out about ##f## and ##g## if we suppose they exist: ##f(x...
  44. thegirl

    Finding the limit of lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1

    Hi, I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit? lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo w and wo are both two points within the same plane.
    • thegirl
    • Thread
    • exponential functions limits maths
    • Replies: 4
    • Forum: Calculus
  45. 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)...
  46. 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...
    • Titan97
    • Thread
    • continuity differentiability functions
    • Replies: 11
    • Forum: General Math
  47. 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...
  48. Mr Davis 97

    What is computation, really?

    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...
  49. 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...
  50. A

    Transforming functions

    Homework Statement If f(x)=|x-1/2|-5 determine g(x)=2f(-x+(3/2)) Homework Equations The Attempt at a Solution Well, I tried to factor out the k-value in the g(x) formula. So I was left with: g(x)=2f(-1)(x-3/2) Then I multiply f(x) by 2 and am left with: g(x)=2|x-(1/2)|-10 Then I subtract...
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