How do you prove PI is the period of tan(x)?
tan(x + T) = sin(x + T) / cos(x + T) = (sin(x)cos(T) + cos(x)sin(T)) / (cos(x)cos(T) - sin(x)sin(T)) For this to equal sin(x) / cos(x) we need sin(T) = 0, cos(t) = ±1. That occurs when x = 0, π, 2π, etc. The smallest positive of these is x = π. Therefore tan(x) = tan(x + π) and so tangent is periodic in π.
How can I prove sin{x} is periodic?
Firstly you should be very clear with these pointsSine is not a equation with which you can prove mathematically, it is a trigonometric function or in simpler terms ratio of the length of perpendicular to length of hypotenuse of a right angled triangle.A function f is said to be periodic with period P (P being a nonzero constant) if we have[math]{\displaystyle f(x+P)=f(x)}[/math]for all values of x in the domain. If there exists a least positive constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tesselations of the plane.So as you can see after every 2π interval, the y co-ordinate of the curve remains same, so thereby we declare that sine function is periodic and has a period of 2π.Hope this helps.Peace :)
If the Function f(x) = Sin(x) + Cos(ax) is periodic, then prove that a is Rational ?
ok..the requirement is that f(x) must be periodic right. f(x) can be periodic only if the 2 functions making it up are periodic (which is true now..).It's period will be the LCM of the periods of the 2 composite(sin(X) and cos(aX))signals.LCM of 2 numbers is always rational. Therefore, sin(x) with period 2*pi and cos(ax) having period=2*pi/a have LCM = 2*pi.Therefore a has to be rational.
Why is the period of tan x pie?
The period of tan x is Pi Then the period of tan (x*pi) is 1 Answer:1
What is the period of tan x?
as we can see from graphTan X= Tan (X+π) = Tan (X+2π) = Tan (X+3π)..so fundamental period of Tan X is π.
What is the period of the function f: R -> R, f(x) = tan(x/3)? Is there a way to know the period without drawing the function?
Note that the period of [math]\tan x[/math] is [math]\pi[/math].Then for the function [math]f:\mathbb R \to \mathbb R, f(x)=\tan (wx)[/math], we have [math]T=\frac{\pi}{w}[/math].When [math]w=\frac{1}{3}[/math], it is clearly that [math]T=\frac{\pi}{\frac{1}{3}}=3\pi[/math].
Prove that: 4 arctan(1/5) - arctan(1/239) = pi/4?
let x = arctan A tan x = A tan (x + x) = (tan x + tan x)/(1 - tan x . tan x) tan 2x = (2 tan x) / (1 - tan^2 x) tan 2x = (2A) / (1 - A^2) 2x = arctan ((2A) / (1 - A^2)) 2 arctan A = arctan ((2A) / (1 - A^2)) 4 arctan A = 2 arctan ((2A) / (1 - A^2)) 4 arctan (1/5) = 2 arctan ((2.1/5) / (1 - 1/25)) = 2 arctan (5/12) = 2 arctan (5/12) - arctan (1/239) = arctan ((2.5/12) / (1 - 25/144)) - arctan (1/239) = arctan (120/119) - arctan (1/239) = arctan ((120/119 - 1/239) / (1 + 120/119 * 1/239) = arctan (1.0042 / 1.0042) = arctan (1) = π/4
Is the square of cos (2t-pi /3) a periodic signal?
YesIn general if f(x) is periodic then any function which takes f(x) only as input will be periodicTherefore (f(x))^2 will also be periodic.For your case cos(2t-π/3) is periodic with period π therefore cos^2(2t-π/3) will also be periodic with same period.Note: fundamental period can differ;actually in this case fundamental period is π/2