Final Answer:
The angles not in the domain of the tangent function on the interval [0, 2π) are θ = π/2, 3π/2, and any angle of the form (2n+1)π/2, where n is an integer.
Step-by-step explanation:
The tangent function, denoted as \( f(\theta) = \tan(\theta) \), is not defined for certain angles due to the nature of its mathematical definition. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. In trigonometry, the tangent function becomes undefined when the adjacent side has a length of zero, as division by zero is undefined in mathematics.
Considering the interval \([0, 2\pi)\), which represents one full revolution around a circle, there are specific angles where the tangent function is not defined. These angles correspond to the values of \( \theta \) that make the adjacent side equal to zero. In trigonometry, this occurs when \( \theta \) is an odd multiple of \( \frac{\pi}{2} \), since at these angles, the cosine of \( \theta \) becomes zero, resulting in a division by zero when calculating the tangent.
Therefore, the angles that are not in the domain of the tangent function on the interval \([0, 2\pi)\) are \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), and any angle of the form \( \frac{(2n+1)\pi}{2} \), where \( n \) is an integer. These angles represent the points in the interval where the tangent function is undefined due to division by zero, and as a result, they should be excluded from the domain when considering the tangent function on this interval.