Introduction to period of tangent function:
In this article, we are going to discuss about determine the period of tangent function. If all real numbers of x, the number of trigonometric function at (t + a) is equal to the number of function at x, that is f(t+a) = f(t) then we can call the function is periodic and the period is 'a'. Now we are going to see some example problems related on the period of trigonometric tangent functions.
Explanation - Period of Tangent Function :
Definition:
The tangent function `tan x` is said to be a periodic function with period `beta` if `tan(x+beta)=tan x` . The least positive value of `beta` is known as the fundamental period of the function. All the circular functions (trigonometrical functions) are periodic function.
For example,
`tan(x+pi) = tan x; tan(x+2pi)= tan x; tan(x+3pi)=tanx` .
Therefore,
`tan(x+npi) = tan x, n in Z` .
Here `beta = . . . -3pi, -2pi, -pi, 0, pi, 2pi, . . .` . But the fundamental period must be least positive quantity.
Therefore `beta = pi` is the fundamental period of tangent function.
Graph:
The graph of the periods of tangent functions y = f(x) = tan x as shown in below.
The trigonometric function `f(x) = tan x` has period `pi` , because `tan(x + pi) = tan x` .
Thus tangent function `tan` is a periodic function with fundamental period `pi` . Similarly one can prove that the functions `cot x` are also periodic functions with fundamental period `pi` while `sin x, cos x, csc x` and `sec x` are periodic with fundamental period `2pi` .
Formula:
(a). A trigonometric function `f(x) = Asin(Btheta + C) + D` .Therefore,
the periods `= (360^o)/|B| = (2pi)/|B|` .
(This formulas are also used by `cos, sec,` and `csc` )
(b). A trigonometric tangent function `f(x) = Atan(Btheta + C) + D` .Therefore,
the periods `= (180^o)/|B| = (pi)/|B|` .
(This formulas are also used by `cot` ).
LCM rule for periods of trigonometric function:
The periodic trigonometric function f(x) = Ag(x) + Bh(x), when the functions g(x) and h(x) are periodic and having periods, `P_1` and `P_2` . The variable A and B are non-zero real values. Therefore, the period of function f(x) determined from the LCM of `P_1` and `P_2` .
` LCM = ` `(LCM - OF - NUMERATORS)/(HCF - OF - DENOMINATORS)` .
For example, `P_1 = (7pi)/3` , and `P_2 = (4pi)/5` .
Period = LCM = `(28pi)/(1) = 28pi` .
Note:
The LCM rule is not applicable for when the given functions are co-functions of compare to other or when the given functions are even.Is this topic How to Find the Volume of a Cylinder hard for you? Watch out for my coming posts.
Period of Tangent Function - Example :
Determine the periods of the given trigonometric tangent functions.
(a). `f(x) = 3tan(6x)` .
Solution:
Given:
`f(x) = 3tan(6x)` .
To find the periods of trigonometric function:
The trigonometric function `f(x) = tan(x)` of graph performed through a complete cycle when the angle evaluates from `0` to `pi` or equivalently the tangent function f(x) = 3tan(6x) of complete cycle performed when `x` evaluates from `0` to `(pi)/6`, that is
`6x = pi` .
`x = (pi)/6` . [By using the above formula].
Therefore, the required periods of the trigonometric tangent functions `f(x)=3tan(6x)` is `(pi)/6` .
In this article, we are going to discuss about determine the period of tangent function. If all real numbers of x, the number of trigonometric function at (t + a) is equal to the number of function at x, that is f(t+a) = f(t) then we can call the function is periodic and the period is 'a'. Now we are going to see some example problems related on the period of trigonometric tangent functions.
Explanation - Period of Tangent Function :
Definition:
The tangent function `tan x` is said to be a periodic function with period `beta` if `tan(x+beta)=tan x` . The least positive value of `beta` is known as the fundamental period of the function. All the circular functions (trigonometrical functions) are periodic function.
For example,
`tan(x+pi) = tan x; tan(x+2pi)= tan x; tan(x+3pi)=tanx` .
Therefore,
`tan(x+npi) = tan x, n in Z` .
Here `beta = . . . -3pi, -2pi, -pi, 0, pi, 2pi, . . .` . But the fundamental period must be least positive quantity.
Therefore `beta = pi` is the fundamental period of tangent function.
Graph:
The graph of the periods of tangent functions y = f(x) = tan x as shown in below.
The trigonometric function `f(x) = tan x` has period `pi` , because `tan(x + pi) = tan x` .
Thus tangent function `tan` is a periodic function with fundamental period `pi` . Similarly one can prove that the functions `cot x` are also periodic functions with fundamental period `pi` while `sin x, cos x, csc x` and `sec x` are periodic with fundamental period `2pi` .
Formula:
(a). A trigonometric function `f(x) = Asin(Btheta + C) + D` .Therefore,
the periods `= (360^o)/|B| = (2pi)/|B|` .
(This formulas are also used by `cos, sec,` and `csc` )
(b). A trigonometric tangent function `f(x) = Atan(Btheta + C) + D` .Therefore,
the periods `= (180^o)/|B| = (pi)/|B|` .
(This formulas are also used by `cot` ).
LCM rule for periods of trigonometric function:
The periodic trigonometric function f(x) = Ag(x) + Bh(x), when the functions g(x) and h(x) are periodic and having periods, `P_1` and `P_2` . The variable A and B are non-zero real values. Therefore, the period of function f(x) determined from the LCM of `P_1` and `P_2` .
` LCM = ` `(LCM - OF - NUMERATORS)/(HCF - OF - DENOMINATORS)` .
For example, `P_1 = (7pi)/3` , and `P_2 = (4pi)/5` .
Period = LCM = `(28pi)/(1) = 28pi` .
Note:
The LCM rule is not applicable for when the given functions are co-functions of compare to other or when the given functions are even.Is this topic How to Find the Volume of a Cylinder hard for you? Watch out for my coming posts.
Period of Tangent Function - Example :
Determine the periods of the given trigonometric tangent functions.
(a). `f(x) = 3tan(6x)` .
Solution:
Given:
`f(x) = 3tan(6x)` .
To find the periods of trigonometric function:
The trigonometric function `f(x) = tan(x)` of graph performed through a complete cycle when the angle evaluates from `0` to `pi` or equivalently the tangent function f(x) = 3tan(6x) of complete cycle performed when `x` evaluates from `0` to `(pi)/6`, that is
`6x = pi` .
`x = (pi)/6` . [By using the above formula].
Therefore, the required periods of the trigonometric tangent functions `f(x)=3tan(6x)` is `(pi)/6` .
No comments:
Post a Comment