Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 30
Find the period and graph the function. $$y=\cot \frac{\pi}{2} x$$
Short Answer
Expert verified
The period of \(y = \cot(\frac{\pi}{2} x)\) is 2.
Step by step solution
01
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.
02
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.
03
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 \(\frac{\pi}{\frac{\pi}{2}} = 2\).
04
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.
05
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.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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: \( \frac{\pi}{\frac{\pi}{2}} = 2 \)
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.
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.
