What is the vertical asymptote of y=tan(x-pi/3)?
You know that tan(π/2) is undefined so there is a vertical asymptote at x=π/2
then x-π/3 = π/2
6x - 2π = 3π
x = 5π/6
there are vertical asymptotes at
x = 5π/6 + kπ, where k is an integer.
To find the vertical asymptote of the function y = tan(x - π/3), we need to analyze the behavior of the tangent function.
The tangent function has vertical asymptotes where the cosine function, which is the denominator of tangent, equals zero. In other words, the vertical asymptotes occur when cos(x - π/3) = 0.
To solve this equation, we set the argument of the cosine function equal to π/2, since the cosine function has a zero at π/2:
x - π/3 = π/2
Next, we solve for x:
x = π/2 + π/3
Simplifying further:
x = (3π + 2π) / 6
x = 5π / 6
Therefore, the vertical asymptote of y = tan(x - π/3) is x = 5π / 6.
To find the vertical asymptote of the function y = tan(x - π/3), we can start by understanding the properties of the tangent function.
The tangent function, denoted as tan(x), has vertical asymptotes at x = (π/2) + πk and x = -(π/2) + πk, where k is an integer.
In this case, the function y = tan(x - π/3) has a horizontal shift of π/3 units to the right compared to the standard tangent function.
Therefore, to find the vertical asymptote of y = tan(x - π/3), we need to find the x-values that satisfy the vertical asymptote condition for the standard tangent function.
For the standard tangent function, the vertical asymptotes occur at x = (π/2) + πk and x = -(π/2) + πk, where k is an integer.
However, due to the horizontal shift of π/3 units to the right in the given function y = tan(x - π/3), we need to replace "x" in the equation of the vertical asymptote with "(x - π/3)".
So, the vertical asymptotes of y = tan(x - π/3) occur at:
x = (π/2) + πk - π/3, and
x = -(π/2) + πk - π/3, where k is an integer.
Simplifying these expressions, we get the vertical asymptotes of the given function as:
x = π/6 + πk, and
x = -5π/6 + πk, where k is an integer.