f(x) = tan x
f'(x) = sec^2 x
Function notation f(x) = tan x represents the tangent function of x, where x is the input angle given in radians. The output of the function is the ratio of the sine of x to the cosine of x, also known as the tangent of x. Mathematically,
f(x) = tan(x) = sin(x) / cos(x)
The tangent function is periodic, meaning it repeats itself every pi (π) radians, or every 180 degrees. It has an infinite set of real solutions with vertical asymptotes at x = (π/2) + nπ, where n is an integer. Additionally, the function has horizontal asymptotes at y = +∞ and y = -∞.
The derivative of the tangent function is given by:
f'(x) = sec^2(x)
Where sec(x) is the secant function of x, defined as the reciprocal of the cosine function.
The range of the tangent function is (-∞, +∞), except at the vertical asymptotes, where the function is not defined. The function takes negative values in intervals (nπ, (n+1)π) for odd values of n, and positive values in intervals (nπ, (n+1)π) for even values of n.
In conclusion, the function f(x) = tan(x) is an important trigonometric function with numerous real-world applications in physics, engineering, and mathematics. Its graph has vertical asymptotes with a period of π, and it is differentiable with a derivative of sec^2(x). The tangent function has a range of (-∞, +∞), except for the intervals where it is undefined.
More Answers:
Understanding the Cosecant Function: Periodicity, Range, Symmetry, Asymptotes, and Zeros.Exploring the Properties of the Secant Function in Trigonometry
Cotangent Function Explained: Evaluating Trigonometric ratio using Cosine and Sine Values