The tangent function, denoted as tan(θ), is undefined when the cosine of the angle θ is zero. This occurs at specific angles where the cosine value equals zero. In the unit circle, these angles are:
- 90 degrees (π/2 radians)
- 270 degrees (3π/2 radians)
- and any angle that is an odd multiple of 90 degrees (or π/2 radians).
At these angles, the tangent function approaches infinity or negative infinity, making it undefined. This is because the tangent of an angle is defined as the ratio of the sine to the cosine of that angle (tan(θ) = sin(θ)/cos(θ)). When the cosine is zero, the denominator becomes zero, leading to an undefined value.
For example, tan(90°) is undefined because cos(90°) = 0, and division by zero is not possible in mathematics.