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.
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
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...
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...
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!
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!
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...
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...
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...
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...
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...
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...
(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...
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...
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...
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...
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...
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"...
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...
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
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}$$
I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...
Hi, I've got this:
$$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close).
Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}##
But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...
I have a set of variables that are always inputs for several functions that I made. Does MatLab have a kind of function that stores these variables into a single matrix (or similar) so that I just need to call this matrix for each function rather than calling them one-by-one as inputs into the...
Hello everybody,
I'm currently helping a friend on an assignment of his, but we are both stumbled on this exercise, I'm posting it here
We define a function ##f## which goes from ##\mathbb{R}## to ##\mathbb{R}## such that its argument maps as
$$
x \mapsto...
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...
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...
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...
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...
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?
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?
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...
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...
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...
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.
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 -...
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...
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 =...
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 ?
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...
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...
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...
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...
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.
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)...
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...
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...
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...
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...
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...