functions Definition and Topics - 56 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. 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...
2. 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!
3. 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!
4. 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...
5. 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...
6. 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...
7. 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...

24. 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...
25. 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...
26. 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...
27. 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...
28. 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?
29. 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?
30. 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...