91Ó°ÊÓ

The tangent function, often denoted as \( \tan(x) \), is one of the six fundamental trigonometric functions and can be expressed as the ratio of the sine function to the cosine function: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function is periodic, meaning it repeats its values in a regular interval. Specifically for tangent, this period is \( \pi \), which is approximately 3.14 radians, or 180 degrees.

Unlike sine and cosine functions, which range between -1 and 1, the tangent function can take on all real numbers, spanning from \(-\infty\) to \(\infty\). This is because the cosine function, found in the denominator of the tangent function, can be zero, leading to values approaching infinity or negative infinity.

The graph of the tangent function has vertical asymptotes, lines which the graph approaches but never touches, at odd multiples of \( \frac{\pi}{2} \) (such as \( \frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2} \), etc.), where the function is undefined. These characteristics of the tangent function significantly impact how we solve equations involving \( \tan(x) \).

For example, when solving \( \tan(x) = 1 \), one finds that the value repeats every \( \pi \) because of this periodic nature. Understanding these properties is crucial when finding solutions to trigonometric equations.

The principal solution of an equation such as \( \tan(x) = 1 \) is the most basic solution or the smallest positive angle that satisfies the equation. In trigonometry, a principal solution is important as it serves as a building block to find more general solutions.

In our original exercise, the principal solution for \( \tan(x) = 1 \) is \( x = \frac{\pi}{4} \), or equivalently, 45 degrees. This is because \( \tan(\frac{\pi}{4}) = 1 \), and \( \frac{\pi}{4} \) is the first positive angle in the first quadrant that satisfies the condition. Principal solutions are often found using reference angles, usually within a range from 0 to \( \pi \) for tangent, ensuring one finds the simplest form of the solution.

Grasping principal solutions helps students tackle more complex equations, as they provide a starting point. From this point, one can explore the periodic nature of the trigonometric function to find all possible solutions.

The general solution of a trigonometric equation takes into account the periodic nature of trigonometric functions. Once you have a principal solution, you can use it to express all possible solutions for the equation. This is done by adding the period of the function (for tangent, it's \( \pi \)) multiplied by an integer \( k \) to the principal solution.

For the equation \( \tan(x) = 1 \), with a principal solution of \( x = \frac{\pi}{4} \), the general solution is given by

• \( x = \frac{\pi}{4} + k\pi \)

where \( k \) can be any integer (0, 1, -1, 2, -2, etc.), allowing us to describe an infinite series of solutions. Each value of \( x \) that satisfies the equation can be reached by choosing different integers for \( k \).

This concept is essential for solving trigonometric equations over different intervals, as it illustrates all the angles that angle \( x \) can take beyond the principal solution. Understanding how to derive a general solution enables students to solve equations across any given range, enhancing their trigonometric problem-solving skills.