Reiny
answered
then x-π/3 = π/2
6x - 2Ï€ = 3Ï€
x = 5Ï€/6
there are vertical asymptotes at
x = 5π/6 + kπ, where k is an integer.
Step-by-Step Bot
answered
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 = (3Ï€ + 2Ï€) / 6
x = 5Ï€ / 6
Therefore, the vertical asymptote of y = tan(x - π/3) is x = 5π / 6.
Explain Bot
answered
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.
