Find the domain of the function y = tan x

The function \( y = \tan x \) is defined for all real numbers except where the cosine function equals zero. This occurs at specific points on the unit circle.

Identifying Restrictions

The tangent function can be expressed as:

\( \tan x = \frac{\sin x}{\cos x} \)

Since division by zero is undefined, we need to find where \( \cos x = 0 \).

Key Points of Discontinuity

• The cosine function is zero at odd multiples of \( \frac{\pi}{2} \), specifically:
• \( x = \frac{\pi}{2} + k\pi \) where \( k \) is any integer.

Domain of the Function

Thus, the domain of \( y = \tan x \) can be expressed as:

\( x \in \mathbb{R} \) except for \( x = \frac{\pi}{2} + k\pi \) where \( k \) is any integer.

In Summary

The function \( y = \tan x \) is defined for all real numbers except at the points where \( \cos x = 0 \). This means the domain excludes odd multiples of \( \frac{\pi}{2} \).