Problem 85 - Precalculus

Question

What is the period of the function \(y=\tan x+\cot x ?\) Use a graphing calculator to graph \(Y_{1}=\tan x+\cot x\) and \(Y_{3}=2 \csc (2 x)\) in the same viewing window.

Step-by-Step Solution

Answer

The period of the function \( y = \tan x + \cot x \) is \( \pi \).

Step 1: Identify the Periods of Individual Functions

The function \(y = \tan x\) has a period of \(\pi\), and the function \(y = \cot x\) also has a period of \(\pi\). Since both functions have the same period, their sum \(y = \tan x + \cot x\) also has a period that is a common multiple of \(\pi\), which is \(\pi\) itself.

Step 2: Confirm with Graphical Observation

Using a graphing calculator, plot \(Y_{1} = \tan x + \cot x\). Observe the graph over the interval \([0, 2\pi]\) or greater to ensure repetitiveness at \(\pi\). The graph should repeat its pattern after every \(\pi\) units, confirming the period is \(\pi\).

Step 3: Compare with Another Function

Plot \(Y_{3} = 2 \csc(2x)\) in the same viewing window. Notice any similarities or differences in periodic behavior compared to \(Y_{1} = \tan x + \cot x\). While both functions will have repeating patterns, their shapes and periods differ significantly, as the period of \(2 \csc(2x)\) is \(\pi\), whereas each component contributes to the general behavior but does not affect the period comparison.

Key Concepts

Periodicity

Periodicity in trigonometric functions refers to the interval at which the function repeats its values. This quality is fundamental to understanding how these functions behave over different intervals. For example, the function \( y = \tan x \) completes one full cycle every \( \pi \) units. This means its graph repeats its pattern every \( \pi \) radians. The same holds true for the function \( y = \cot x \).

When functions like \( \tan x \) and \( \cot x \) are combined, their periods must be considered to find the new period of their sum. Since both have the same period of \( \pi \), their sum \( y = \tan x + \cot x \) also has a period of \( \pi \). This makes the sum highly predictable and manageable, demonstrating how periodicity remains intact even through combinations.

Tan Function

The tangent function, \( \tan x \), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). This function has some distinct characteristics:

• Periodicity: Its period is \( \pi \), meaning it repeats every \( \pi \) radians.
• Vertical Asymptotes: These occur at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer, causing \( \cos x = 0 \).
• Odd Function: Due to the property \( \tan(-x) = -\tan x \).

The repetitive nature of \( \tan x \) ensures that understanding a single period \( [0, \pi) \) helps predict its behavior indefinitely. Recognizing these features can aid in sketching its graph or understanding it in combination with other functions.

Cotangent Function

The cotangent function, \( \cot x \), complements the tangent function by providing another perspective on ratios of sides in a right triangle. Defined as \( \cot x = \frac{\cos x}{\sin x} \), it shares some similarities and differences with tangent. Key characteristics include:

• Periodicity: With a period of \( \pi \), identical to that of \( \tan x \).
• Horizontal Mirrors: \( \cot x \'s \) range between its vertical asymptotes go from \( +\infty \) to \(-\infty \).
• Vertical Asymptotes: Present at \( x = k\pi \), with \( k \) as any integer, due to \( \sin x = 0 \).

The periodic nature and unique asymptotic properties differentiate \( \cot x \) from other trigonometric functions but align in a predictable pattern with \( \tan x \). This congruence ensures that when additively combined, as in \( \tan x + \cot x \), the period remains \( \pi \).

Graphical Analysis

Graphical analysis involves visually interpreting the behavior and properties of trigonometric functions. Through graphing software or calculators, we can plot functions like \( y = \tan x + \cot x \) to observe their periodicity and overall behavior. When graphing:

- Plot \( Y_{1} = \tan x + \cot x \) over an interval such as \([0, 2\pi]\) to see the repeating pattern every \( \pi \) units.- Compare it with another function, like \( Y_{3} = 2 \csc(2x) \), in the same window to contrast periods. Both will show periodic behavior but differ in specifics.
Visual analysis provides quick insights into the mathematical properties of the functions. Looking at a graph helps confirm the period and identify any unique characteristics such as asymptotes or amplitude changes. This visual approach is particularly valuable for understanding complex combinations like \( \tan x + \cot x \).