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The tangent function, denoted as \( \tan{x} \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine of an angle to the cosine of that angle: \( \tan{x} = \frac{\sin{x}}{\cos{x}} \). The tangent function is periodic, with a period of \( \pi \), meaning it repeats its values every \( \pi \) units. This characteristic is crucial for solving equations involving \( \tan \), as solutions recur at regular intervals.

When solving equations such as \( \tan x = 1 \), it's important to recall where the tangent equals this value. The tangent function equals 1 at an angle of \( \frac{\pi}{4} \), as well as at every integer multiple of its period, \( n\pi \), added to \( \frac{\pi}{4} \). Thus, the general solution is written as \( x = \frac{\pi}{4} + n\pi \), where \( n \) is any integer.

This regular pattern helps us find all possible solutions within any specified interval, such as \([-2\pi, 2\pi]\). In this specific exercise, the solutions are \(-\frac{7\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}\).

Using a graph to solve trigonometric equations provides a visual understanding of where solutions lie within a given interval. The graph of the tangent function has a distinctive repeating pattern, with vertical asymptotes occurring where the cosine function is zero, typically at odd multiples of \( \frac{\pi}{2} \). These asymptotes are crucial to note because the tangent function becomes undefined at these points.

By plotting \( y = \tan x \) and \( y = 1 \) on the same graph, we can visually identify solution points as the intersections of the two curves. In the interval \([-2\pi, 2\pi]\), these intersection points are the values of \( x \) where \( \tan x = 1 \) occurs, confirming our analytical approach.

• At \( x = -\frac{7\pi}{4} \), the graph of \( \tan x \) crosses \( y = 1 \).
• Similarly, at \( x = -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4} \), the graph coincides with \( y = 1 \).

Graphical solutions complement algebraic methods by providing a clear picture and validating analytical solutions.

Interval notation is a way of representing a range of values, often used in mathematics to specify the domain or range of solutions. The interval \([-2\pi, 2\pi]\) is a closed interval, indicated by the square brackets, meaning it includes the endpoints \(-2\pi\) and \(2\pi\). This notation is especially useful when dealing with continuous functions like \( \tan x \), where we need to specify the exact segment of the number line under consideration.

When solving trigonometric equations, it's crucial to interpret the interval correctly to determine all valid solutions within it. This means choosing integer values for \( n \) in the general solution of \( x = \frac{\pi}{4} + n\pi \) such that the resulting x-values remain within \([-2\pi, 2\pi]\).

Using interval notation helps streamline the solution process by clearly defining the boundaries and ensuring all solutions fall within those limits. It provides a concise and universal way to denote specific ranges, making it easier to communicate and verify solutions.