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The tangent function, denoted as \( \tan z \), is one of the three primary trigonometric functions, alongside sine and cosine. It is the ratio of the sine of an angle to the cosine of that angle, expressed as:

• \( \tan z = \frac{\sin z}{\cos z} \)

The tangent function is defined for all real numbers except for values where the cosine is zero, which occurs at odd multiples of \( \frac{\pi}{2} \). This is because division by zero is undefined.
The properties of the tangent - being periodic with period \( \pi \) - mean its graph repeats every \( \pi \) units. Thus, evaluating \( \tan(z) \) within one period \( [0, \pi) \) can help predict its behavior across all real numbers.
Understanding these characteristics aids in solving equations and determining outputs for given angles.

The range of the tangent function is particularly interesting due to its expansive nature. Unlike sine and cosine, which have restricted ranges, the tangent function extends to all real numbers. In more formal terms, we describe this as:

• The range of \( \tan z \) is all real numbers \( \mathbb{R} \).

Therefore, for any real number you can think of —be it positive, negative, large, or small— there exists some angle \( z \) for which the tangent of that angle equals this number.
It makes the tangent function very inclusive, as no real number is excluded from its range. This unique trait is crucial for confirming whether a value like \( \frac{3 \pi}{2} \) can be obtained from the tangent of some angle.
Thus, even uncommon or complex values like \( \frac{3\pi}{2} \) are achievable as outputs of the tangent function.

Real numbers encompass an extensive set of numbers. They include:

• Natural numbers (1, 2, 3, ...)
• Whole numbers (0, 1, 2, ...)
• Integers (..., -2, -1, 0, 1, 2, ...)
• Rational numbers (fractions like \( \frac{1}{2}, \frac{2}{3} \))
• Irrational numbers (such as \( \sqrt{2} \) and \( \pi \))

This extensive set is symbolized by \( \mathbb{R} \). When the problem asks if \( \tan z = \frac{3\pi}{2} \) is possible for some real number \( z \), it is essentially asking if \( \frac{3\pi}{2} \) fits within this vast pool.
Because the real numbers include any conceivable number that could be tangent output, confirming this is straightforward. The statement highlights the true scope of the real number system and its applicability across various functions, including trigonometric ones.
Such understanding is valuable for seeing how mathematics operates beyond mere integers and simple fractions.