Solving tanx 0 in the Interval -180° ≤ x ≤ 180°: A Comprehensive Guide

Introduction to Trigonometry and the Tangent Function

In trigonometry, the tangent function is one of the fundamental trigonometric functions, denoted as tan(x). It is defined as the ratio of the sine of an angle to the cosine of the same angle. Understanding the tangent function is crucial for solving various trigonometric problems.

Identifying When tanx 0

The equation tanx 0 indicates that the tangent of an angle is zero. To solve this, we need to identify the angles where the tangent function equals zero. The tangent function is a periodic function and is zero at integer multiples of π radians, or equivalently, at integer multiples of 180 degrees.

General Solution

The general solution to the equation tanx 0 is given by:

For any integer n: tan(nπ) 0

This means that the tangent of nπ radians (or n times; 180°) will always be zero.

Applying the Interval -180° ≤ x ≤ 180°

Given the interval -180° ≤ x ≤ 180°, we need to find the specific integer values of n that satisfy the equation tanx 0 within this range.

For n -1: x -1 times; 180° -180° For n 0: x 0 times; 180° 0° For n 1: x 1 times; 180° 180°

These are the only integer multiples of 180 degrees within the given interval. Therefore, the solutions for tanx 0 in the interval -180° ≤ x ≤ 180° are:

x -180°, 0°, 180°

It is important to note that while the tangent function approaches zero as it gets closer to 180° or -180°, these specific points are exactly where the tangent function equals zero.

Further Exploration and Applications

Understanding the tangent function and its zeros is fundamental in various fields, including engineering, physics, and mathematics. This knowledge can be applied in solving more complex trigonometric equations and in real-world scenarios where trigonometric functions are used to model periodic behavior.

For instance, in physics, the tangent function is used to analyze the behavior of waves or oscillations, and in engineering, it helps in analyzing signals and systems. In mathematics, the tangent function is crucial in calculus, where it is used in the study of derivatives and integrals.

Conclusion

In summary, solving the equation tanx 0 within the interval -180° ≤ x ≤ 180° involves identifying the integer multiples of 180 degrees that fall within this range. The solutions are -180°, 0°, and 180°. This understanding is not only crucial in trigonometry but also in various practical applications across disciplines.

We hope this guide helps you in mastering the concept of the tangent function and its zeros in the given interval. For further exploration, consider delving into more advanced topics in trigonometry and related fields.