The zeros of the tangent function, tan(x), occur at x = nπ, where n is any integer (…,-2, -1, 0, 1, 2,…). This means the tangent function equals zero at every multiple of π (approximately 3.14159).
Understanding the Tangent Function
The tangent function is defined as the ratio of sine to cosine: tan(x) = sin(x) / cos(x). Therefore, tan(x) will equal zero whenever sin(x) = 0, and cos(x) ≠ 0. The sine function, sin(x), is zero at every multiple of π.
- Example:
- tan(0) = sin(0) / cos(0) = 0 / 1 = 0
- tan(π) = sin(π) / cos(π) = 0 / -1 = 0
- tan(2π) = sin(2π) / cos(2π) = 0 / 1 = 0
- and so on...
Visual Representation
The graph of y = tan(x) visually confirms this. The function crosses the x-axis (y=0) at every multiple of π. Note that the tangent function has vertical asymptotes where cos(x) = 0, which occurs at odd multiples of π/2. This is because division by zero is undefined.
Practical Applications
Understanding the zeros of the tangent function is crucial in various applications, including:
- Solving Trigonometric Equations: Finding solutions to equations involving the tangent function.
- Calculus: Analyzing the behavior of functions involving the tangent function, such as finding derivatives and integrals.
- Physics and Engineering: Modeling periodic phenomena where the tangent function plays a role, such as oscillations and waves.
The provided references confirm that tan(0°) = 0 and that the zeros of tan(x) are at x = nπ, where n is an integer. The Mathway link, while not directly providing the answer, points to resources that solve trigonometric problems. The Stack Exchange link explicitly states that zeros exist when sin(x) = 0, which occurs at x = nπ. Other links provide supporting information on related trigonometric concepts.
