Problem 1 - Precalculus Mathematics for Calculus

Question

The trigonometric function \(y=\tan x\) has period _______ and asymptotes \(x=\) _______. Sketch a graph of this function the interval \((-\pi / 2, \pi / 2)\)

Step-by-Step Solution

Answer

The period is \(\pi\); asymptotes are at \(x = \frac{\pi}{2} + k\pi\).

Step 1: Understanding the Periodicity of \(\tan x\)

The function \(y = \tan x\) is periodic, repeating itself every \(\pi\) radians. This means that the period of the tangent function is \(\pi\).

Step 2: Identifying the Asymptotes

The function \(y = \tan x\) has vertical asymptotes at the points where \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. Therefore, in the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), there are asymptotes at \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\).

Step 3: Sketching the Graph in the Given Interval

In the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), the graph of \(y = \tan x\) starts just below the asymptote at \(x = -\frac{\pi}{2}\), crosses the x-axis at the origin, and approaches the asymptote at \(x = \frac{\pi}{2}\) from below, without ever touching it. The function increases continuously from \(-\infty\) to \(+\infty\) as it moves across this domain.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and ratios of right triangles. They are fundamental in understanding various phenomena in both natural and social sciences. The primary trigonometric functions include the sine, cosine, and tangent.
For the tangent function specifically, it is defined as the ratio of the sine of an angle to the cosine of the same angle:

• \[\tan x = \frac{\sin x}{\cos x}\]

The function represents how steep a line is that can be drawn from the origin at any given angle \(x\). As the cosine of the angle approaches zero, the tangent of the angle approaches \(+\infty\) or \(-\infty\), and this is where vertical asymptotes occur. Understanding the basic properties of the tangent function is key to mastering its behavior and applications.

Periodicity of Functions

Periodicity is an essential trait of trigonometric functions. It refers to how these functions repeat their values at regular intervals. For the tangent function, its periodicity means its pattern repeats every \(\pi\) radians. This is shorter than sine and cosine functions, which repeat every \(2\pi\) radians.
The periodic nature of \(\tan x\) allows us to predict how its graph looks beyond basic intervals. In a mathematical sense, if a function is periodic with period \(T\), then:

• \[f(x + T) = f(x)\]

For tangent specifically, this property is both a challenge and an opportunity since it ensures continuity and repetition in its pattern. Recognizing this periodicity in tangent and other trigonometric functions simplifies solving equations and graphing.

Vertical Asymptotes

Vertical asymptotes are crucial in understanding the graph behavior of the tangent function. These asymptotes indicate points where the function sharply rises or falls, approached by the graph but never actually crossed or touched.
For \(y = \tan x\), vertical asymptotes occur at positions where the cosine function is zero because the tangent is undefined at those points:

• At every \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer, the vertical asymptotes appear.

This means within a single period of the tangent function, from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the function will have an asymptote at each end:

• \(x = -\frac{\pi}{2}\)
• \(x = \frac{\pi}{2}\)

Students must identify these asymptotes accurately to understand where the tangent function breaks and how it behaves beyond these points. The steep climb towards the asymptotes is a characteristic feature of the tangent graph, underscoring its complex and fascinating nature.