Problem 30 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\cot \frac{\pi}{2} x$$

Step-by-Step Solution

Answer

The period of \(y = \cot(\frac{\pi}{2} x)\) is 2.

Step 1: Understanding the Cotangent Function

The cotangent function, denoted as \(\cot(x)\), is the reciprocal of the tangent function, or \(\cot(x) = \frac{1}{\tan(x)}\). The basic period for \(\cot(x)\) is \(\pi\). Our task is to determine how the given function differs from the standard form.

Step 2: Identify the Argument

In the given function \(y = \cot\left(\frac{\pi}{2} x\right)\), the argument of the cotangent function is \(\frac{\pi}{2}x\). This modifies the period compared to the basic cotangent function.

Step 3: Calculating the Period

The period of \(y = \cot\left(k x\right)\) is calculated as \(\frac{\pi}{|k|}\). Here, \(k = \frac{\pi}{2}\). Thus, the period is \(\frac{\pi}{\frac{\pi}{2}} = 2\).

Step 4: Graphing Preparation

With the period determined to be 2, we note that the cotangent function begins at zero and undefined values at multiples of its period \([n, n+2]\) where \(n\) is an integer. The function decreases as it approaches these undefined points.

Step 5: Sketch the Graph

To graph \(y = \cot\left(\frac{\pi}{2} x\right)\), plot key points and vertical asymptotes, which occur at \(x = 0, 2, 4, ...\) and show a complete cycle between each \(n\) and \(n+2\). Draw a smooth curve starting at zero, decreasing as it approaches each vertical asymptote, and this pattern repeats every 2 units.

Key Concepts

Period Calculation

Calculating the period of a trigonometric function is a vital step in understanding its behavior. The period of a function refers to the length of the interval over which it repeats itself. For the cotangent function with an argument such as \(y = \cot\left(\frac{\pi}{2}x\right)\), we use the formula \( \frac{\pi}{|k|} \) to find the period. Here, \(k\) represents the coefficient of \(x\) within the cotangent argument. In our example, \(k = \frac{\pi}{2}\), hence

• Substitute into the period formula: \( \frac{\pi}{\frac{\pi}{2}} \)
• Simplify the expression: \( \frac{\pi}{\frac{\pi}{2}} = 2 \)

The outcome tells us the function repeats every 2 units along the x-axis. Understanding the modified period, compared to the standard cotangent period of \(\pi\), helps in accurately sketching and analyzing its graph.

Trigonometric Graphing

Graphing a trigonometric function requires plotting several important features. For cotangent functions, the essential points include zeros and vertical asymptotes.

• The cotangent function, \(y = \cotx\), theoretically approaches infinity at multiples of \(\pi\) and zero at mid-points between these values.
• In our function \(y = \cot\left(\frac{\pi}{2}x\right)\), vertical asymptotes occur at \(x = 0, 2, 4, \ldots\)
• Behavior: Between each asymptote, the function decreases from positive infinity to negative infinity.

The graph becomes periodic with a complete cycle every 2 units. Plot key points, including the zeros at regular intervals, to assist in drawing a smooth curve over the repeated cycles. The cotangent graph differs from sine and cosine by having a vertical structure, with distinct decreasing patterns between its asymptotes. By observing these characteristics, one can master the visualization of cotangent graphs.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions form a critical part of trigonometry, allowing us to express relationships between angles and ratios differently. Cotangent, cosecant, and secant are the primary reciprocal functions.

• The cotangent function is expressed as \( \cotx = \frac{1}{\tanx} \), making it infinite wherever tangent is zero and zero where tangent is infinite.
• These functions reflect important properties such as asymptotic behavior and vertical trends, differing from their counterparts.

Understanding cotangent as a reciprocal emphasizes its dependence on tangent and clarifies its transformations, such as period adjustments. Other reciprocal functions follow similar logic, aiding in solving trigonometric equations or transformations. Recognizing how cotangent and related functions invert their base trigonometric partners can provide insights into their applications across various contexts.