Problem 1 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The period is \( \pi \) and asymptotes occur at \(x = \pm \frac{\pi}{2}\).

Step 1: Identify the Period of the Function

The function given is the tangent function, \(y = \tan x\). The period of the tangent function is the interval after which it starts repeating its values. For \(\tan x\), this period is \(\pi\). Therefore, the period of \(y = \tan x\) is \(\pi\).

Step 2: Determine the Asymptotes of the Function

The tangent function has vertical asymptotes where it is undefined, which occur at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\). To find the asymptotes within the given interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), we focus on \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\). Thus the asymptotes in this interval are \(x = \pm \frac{\pi}{2}\).

Step 3: Sketch the Graph

Plot the tangent curve in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\). The graph starts from negative infinity near \(x = -\frac{\pi}{2}\), crosses through the origin (0,0), and approaches positive infinity as it nears \(x = \frac{\pi}{2}\), forming a characteristic S-like shape between the asymptotes \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\).

Key Concepts

Tangent Function

The tangent function, represented as \(y = \tan x\), is an important trigonometric function that emerges from dividing the sine function by the cosine function. It is defined as:

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

This function has unique properties that distinguish it from other trigonometric functions like sine and cosine.
One key characteristic of the tangent function is that it is periodic, meaning it repeats its values over specific intervals.
The graph of \( \tan x \) has a distinct "S" shape, crossing the origin at \((0, 0)\). However, unlike sine and cosine functions, the tangent function does not oscillate between a maximum and minimum value. Instead, it continuously increases or decreases without bound.
When graphing \( \tan x \), you'll notice that it extends from negative to positive infinity within its defined intervals.
This behavior results in certain peculiar elements such as vertical asymptotes and a specific period.

Periodicity

In trigonometry, periodicity refers to the characteristic of a function to repeat its values at regular intervals. For the tangent function \(y=\tan x\), this period is \(\pi\). What this means is that every \(\pi\) units along the x-axis, the graph of \(\tan x\) will start repeating itself.
An easy way to understand this is by considering the graph between two asymptotes, such as \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\). Within this domain, the graph travels from negative infinity, crosses the x-axis at zero, and then climbs to positive infinity as it approaches the next asymptote.

• At \(x = 0\), \(\tan x\) equals zero.
• At \(x = \frac{\pi}{4}\), \(\tan x\) equals 1.
• At \(x = -\frac{\pi}{4}\), \(\tan x\) equals -1.

The periodicity of \(\tan x\) helps define its behavior and is crucial when predicting the function's behavior beyond the initial cycle.

Vertical Asymptotes

Vertical asymptotes are lines that the curve of a function approaches but never actually intersects. For the tangent function \(y = \tan x\), these occur where the function is undefined. This happens whenever the cosine denominator in \(\frac{\sin x}{\cos x}\) equals zero.

Therefore, the points of vertical asymptotes for \(\tan x\) are at:

• \(x = \frac{\pi}{2} + k\pi\)

where \(k\) is any integer. This equation tells us that asymptotes form at every half-period interval of \(\tan x\), starting from \(\pm \frac{\pi}{2}\).
In the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), the asymptotes are found at \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\).

Within these bounded regions, the function value of \(\tan x\) shoots to infinity (positive or negative) as it nears these vertical boundaries until once more approaching zero. Vertical asymptotes are critical as they signal the breakpoints in the function's graph and highlight its undefined range of values.