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In trigonometry, the tangent function is one of the primary functions we encounter. It is often abbreviated as 'tan' and uniquely relates to angles in a right-angled triangle as the ratio of the side opposite to the angle to the side adjacent to it. More formally, for an angle \( \theta \), the tangent function is defined as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

But the tangent function extends beyond the confines of a triangle in a plane. It is represented graphically as a wave that continues indefinitely in both the positive and negative directions of the x-axis, intersecting it at multiples of \( \pi \) radians (180 degrees) and having undefined values (asymptotes) at multiples of \( \frac{\pi}{2} \).

The period of the tangent function is the length of one full cycle of the wave before it starts repeating itself and for \( \tan(x) \), this period is \( \pi \).

Vertical scaling in the context of functions, including trigonometric functions like the tangent, refers to the stretching or compressing of the graph of a function in the vertical direction. This transformation does not affect the period of the function, but rather the amplitude – the height of the peaks and the depth of the troughs.

When we apply vertical scaling to the tangent function, for example, by multiplying it by a constant \( a \), the new function becomes \( y = a \tan(x) \). Although this changes the steepness of the slopes leading to and away from the asymptotes, it does not change the points where the function intersects the x-axis, or the location of the asymptotes themselves. Thus, the fundamental wave shape remains the same, and so does the period.

Trigonometry is a branch of mathematics that deals with triangles, focusing especially on right ones, and the relationships between their angles and sides. The central elements of trigonometry are the trigonometric functions: sine, cosine, and tangent, along with their reciprocals cosecant, secant, and cotangent.

These functions have applications spanning from simple measurements in geometry to more complex problems in calculus, physics, engineering, and astronomy. They are periodic functions, meaning they repeat their values in regular intervals. Understanding how to manipulate and interpret these trigonometric functions is fundamental to solving many mathematical problems, including finding the period of a function, such as in the exercise presented.