Problem 14 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=2 \cot x $$

Step-by-Step Solution

Answer

The period is \( \pi \). Graph the function over \( 0 \) to \( \pi \), applying vertical scaling by 2.

Step 1: Understanding the Function Type

The function given is a trigonometric function, specifically the cotangent function. The cotangent function, denoted as \( \cot x \), is the reciprocal of the tangent function.

Step 2: Determine the Basic Period of \( \cot x \)

The cotangent function has a period of \( \pi \) because \( \cot(x + \pi) = \cot x \). This is because the tangent function has a period of \( \pi \), and cotangent is its reciprocal.

Step 3: Identify Modifications to the Function

Our function is \( y = 2 \cot x \). The '2' is a coefficient that affects the amplitude, but does not affect the period. Thus, the period remains the same as that of \( \cot x \).

Step 4: Determine the Period of \( y = 2 \cot x \)

Since the coefficient in front of \( \cot x \) does not affect the period, the period of \( y = 2 \cot x \) is \( \pi \).

Step 5: Graph the Function

To graph \( y = 2 \cot x \), plot the basic shape of the cotangent function over one period from \( 0 \) to \( \pi \). Keep in mind that \( \cot x \) has vertical asymptotes at \( x = n\pi \), where \( n \) is an integer, and zeros at \( x = (n+0.5)\pi \). Scale the values by the factor of 2 accordingly.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot x \), is a fundamental trigonometric function. It is the reciprocal of the tangent function. This means, mathematically, that \( \cot x = \frac{1}{\tan x} \). The basic idea is that while tangent measures the opposite side over the adjacent side of a right triangle, cotangent measures the adjacent side over the opposite side.
The cotangent function helps us in various mathematical and applied fields, like physics, engineering, and navigation. It provides a unique perspective, often letting us solve equations or model phenomena where tangent might not suffice. Remember:

• The cotangent function is undefined wherever the tangent function is zero. Typically, these are points at multiples of \( \pi \), such as \( 0, \pi, 2\pi, \text{etc.} \).
• Cotangent graphs show vertical asymptotes at these points because the denominator in its fraction form approaches zero.
• Cotangent values are highest at points where tangent values are low since they are inversely related.

Understanding these properties allows you to decipher the behavior and graph of the cotangent function effortlessly.

Period of Trigonometric Functions

A trigonometric function's period is the length of the smallest interval over which the function repeats its values. This is a critical property for functions like sine, cosine, tangent, and cotangent, as many natural and mathematical phenomena exhibit periodic behavior.
For the cotangent function, the period is \( \pi \). This stems from its reciprocal relationship with the tangent function, which also has a period of \( \pi \). Essentially, this means that the cotangent function repeats its pattern every \( \pi \) units.
It's important to note that transforming functions, such as adding a coefficient to \( \cot x \), does not change the period. For example, in the function \( y = 2 \cot x \), the number 2 scales the output values but maintains the same period of \( \pi \). This invariance is central to understanding how modifications can affect trigonometric functions' outputs while leaving their periodic nature untouched.

Graphing Trigonometric Functions

Graphing trigonometric functions allows us to visualize how these functions behave over a cycle or period. Let's consider graphing \( y = 2 \cot x \) as an example.
To start, identify the key features of the cotangent graph. Each period of \( \cot x \) contains a vertical asymptote at each end, with zeros midway through. For \( y = 2 \cot x \), the basic shapes of these graphs will look the same over each \( \pi \) interval, but the outputs will simply be doubled in magnitude.
Here's a step-by-step approach to graph \( y = 2 \cot x \):

• Plot the vertical asymptotes at \( x = n\pi \), with zero values at \( x = (n+0.5)\pi \). This root placement does not change with the coefficient.
• Between each asymptote pair (such as \( 0 \) and \( \pi \)), sketch the cotangent curve, reflecting the negative slope nature of the function.
• Scale the graph by 2 because of the coefficient, stretching the graph vertically.

Understanding these graphing principles helps accurately represent function transformations. Practice with multiple functions to get proficient at interpreting and predicting trigonometric behavior.