Question

What are the equations of the asymptotes for y=tan(x) in the interval 0 < x < 2π?

Answer 1

The function \( y = \tan(x) \) has vertical asymptotes where the function is undefined, which occurs when the cosine function in the denominator (since \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)) is equal to zero.

The cosine function is zero at the odd multiples of \( \frac{\pi}{2} \). Specifically, we can determine the vertical asymptotes by solving the equation:

\[
\cos(x) = 0
\]

This happens at:

\[
x = \frac{\pi}{2} + n\pi \quad \text{for integer } n.
\]

In the interval \( 0 < x < 2\pi \), the values of \( n \) that keep \( x \) within this range are \( n = 0 \) and \( n = 1 \):

1. For \( n = 0 \):
\[
x = \frac{\pi}{2}
\]

2. For \( n = 1 \):
\[
x = \frac{\pi}{2} + \pi = \frac{3\pi}{2}
\]

Thus, the equations of the vertical asymptotes for \( y = \tan(x) \) in the interval \( 0 < x < 2\pi \) are:

\[
x = \frac{\pi}{2} \quad \text{and} \quad x = \frac{3\pi}{2}.
\]