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The tangent function is a fundamental aspect of trigonometry and represents the ratio of the opposite side to the adjacent side in a right-angled triangle. Formally, for a given angle \theta, it is defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

It is periodic with a period of \( \pi \) which means it repeats its values every \( \pi \) radians. The function is undefined when the cosine of the angle is 0, which occurs at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). This also sets the location of its vertical asymptotes.

When solving equations that involve the tangent function, we often seek angles that produce a specific tangent value. In the case of \( \tan x = 18 \) from the example, we start by using an inverse trigonometric function to find the principle solution, which represents the angle whose tangent is 18 within the regular range of \( -\frac{\pi}{2} < x < \frac{\pi}{2} \). But since \( \tan \) is periodic, there are infinitely many solutions which can be found by adding integer multiples of its period.

The arctangent function, denoted as \( \arctan \), is the inverse of the tangent function. Given a value, the arctangent returns the angle whose tangent is that value. This function is vital for solving trigonometric equations where the tangent of an angle is known, and the goal is to determine the possible angles.

For example, in the equation \( \tan x = 18 \), applying the arctangent to both sides yields \( x = \arctan(18) \). It provides us with the principle value, which lies within \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians. A calculator can be used to find the approximate value of \( x \).

Because the tangent function is periodic, the arctangent only accounts for one possible angle within its principal range. To find additional solutions, we consider the periodic nature of the tangent function and seek angles that are congruent modulo the period \( \pi \) radians.

Trigonometric identities are equations involving trigonometric functions that hold true for all allowed values of the variables within the functions. They are essential tools for simplifying trigonometric expressions and solving equations. Some of the basic identities include the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and angle sum and difference identities like \( \sin(\theta \pm \phi) \) and \( \cos(\theta \pm \phi) \).

In the context of solving trigonometric equations, identities allow us to manipulate and rewrite equations in ways that are easier to solve. Understanding these identities is crucial for finding all possible solutions within a given interval, such as \( [0, 2\pi) \) in the provided exercise. By recognizing the periodicity of trigonometric functions, we can effectively use identities to solve complex problems, making them an indispensable part of trigonometry.