Question

Show that the range of the tangent function is the set of all real numbers.

Short answer

The range of the tangent function is all real numbers.

Step-by-step solution

Understand the Tangent Function

The tangent function is defined as \(\tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)}\). It is periodic with a period of \(\frac{\text{π}}{2}\) and has vertical asymptotes where \(\text{cos}(x) = 0\).

Identify Key Properties

The tangent function is continuous and unbounded in the intervals between its vertical asymptotes. This means that it can take on any values from \(-\text{∞}\) to \(\text{+∞}\) as \(x\) varies within its period.

Mathematical Proof Using Limits

To show this formally, consider the limit of \(\tan(x)\) as \(x\) approaches a point where \(\text{cos}(x) = 0\). As \(x\) approaches \(\frac{\text{π}}{2}\) from the left, \(\tan(x)\) approaches \( +\text{∞} \) and as \(x\) approaches \(\frac{\text{π}}{2}\) from the right, \(\tan(x)\) approaches \( -\text{∞} \).

Conclusion

Since the tangent function can take any value between \(-\text{∞}\) and \(+\text{∞}\) within each interval of its period, we can conclude that the range of the tangent function is all real numbers.

Key concepts

Tangent Function

The tangent function, often written as \(\tan(x)\), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function: \[\tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)}\]. This definition means that wherever \(\text{cos}(x)\) equals zero, \(\tan(x)\) is undefined, leading to vertical asymptotes. The tangent function is periodic with a period of \(\text{π}\). This means the pattern of values it takes repeats every \(\text{π}\) units.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of periodic phenomena. The sine function \(\text{sin}(x)\) gives the y-coordinate on the unit circle, and the cosine function \(\text{cos}(x)\) gives the x-coordinate. These functions are periodic with a period of \(\text{2π}\), but the tangent function, derived from the sine and cosine, has a shorter period of \(\text{π}\). These functions are continuously used in applications ranging from physics to engineering.

Limits

In calculus, the concept of limits describes the behavior of a function as its input approaches a certain value. For the tangent function, we look at its limits as it approaches the points where \(\text{cos}(x)\) is zero. For example, as \(\text{x}\) approaches \(\frac{\text{π}}{2}\) from the left, \(\tan(x)\) approaches \(\text{+∞}\), and as it approaches \(\frac{\text{π}}{2}\) from the right, \(\tan(x)\) approaches \(\text{-∞}\). This unbounded behavior is key to understanding the range of the tangent function.

Periodicity

Periodicity refers to the repeating nature of functions. For the tangent function, this property is evident in its period of \(\text{π}\). Every \(\text{π}\) units, the function's values repeat. This periodic behavior means that within each interval of \(\text{π}\) between vertical asymptotes, the function takes all values from \(\text{-∞}\) to \(\text{+∞}\). This periodic unboundedness confirms that the range of the tangent function is the set of all real numbers.

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity as the input approaches a specific value. For the tangent function, vertical asymptotes occur at \(\frac{\text{π}}{2}, \frac{3\text{π}}{2}, \frac{5\text{π}}{2},\) and so on. These points are where \(\text{cos}(x)\) equals zero, making the tangent function undefined. As \(\text{x}\) approaches these points, \(\tan(x)\) increases or decreases without bound. These asymptotes are crucial in verifying that the tangent function can take any real number as its output, contributing to its infinite range.