Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$(\cot x-1)(\sqrt{3} \cot x+1)=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{2\pi}{3}, \frac{5\pi}{3}\).

Step 1: Understand the Given Equation

The given equation is \((\cot x - 1)(\sqrt{3} \cot x + 1) = 0\). This is a product of two factors. According to the zero-product property, if the product of two factors is zero, at least one of the factors must be zero.

Step 2: Set Each Factor to Zero

We will set each factor to zero and solve for \(x\).1. Solve \(\cot x - 1 = 0\)2. Solve \(\sqrt{3} \cot x + 1 = 0\)

Step 3: Solve for x in the First Equation

First, solve \(\cot x - 1 = 0\):\[\cot x = 1\]The cotangent of an angle is 1 when the angle is \(\frac{\pi}{4}\) or \(\frac{5\pi}{4}\), based on the periodicity of \(\cot\). Thus, \(x = \frac{\pi}{4}, \frac{5\pi}{4}\).

Step 4: Solve for x in the Second Equation

Next, solve \(\sqrt{3} \cot x + 1 = 0\):\[\cot x = -\frac{1}{\sqrt{3}}\]The cotangent of an angle is \(-\frac{1}{\sqrt{3}}\) when the angle is \(\frac{2\pi}{3}\) or \(\frac{5\pi}{3}\), considering the cotangent's symmetry and periodicity. Thus, \(x = \frac{2\pi}{3}, \frac{5\pi}{3}\).

Step 5: Combine All Solutions

Compile all the solutions from both equations:\(x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{2\pi}{3}, \frac{5\pi}{3}\).

Key Concepts

Zero-Product Property

In mathematics, the zero-product property is a fundamental principle that helps us solve equations containing products.
It states that if the product of two or more factors equals zero, then at least one of those factors must be zero.
This principle is expressed mathematically as:

• If \( a \times b = 0 \), then \( a = 0 \) or \( b = 0 \).

For trigonometric equations like the one in the given exercise, identifying zeros of each factor helps isolate specific solutions for the variable in question. When dealing with trigonometric functions, setting each factor equal to zero simplifies solving by reducing it to handling individual functions such as sine, cosine, or in this case, cotangent.
Breaking down the process step-by-step as per the zero-product property clarifies the path towards deriving all potential solutions within the specified interval. This logical structure ensures that none of the possible roots of the equation are neglected.

Cotangent Function

The cotangent function, represented as \( \cot x \), is the reciprocal of the tangent function.
Mathematically, \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).
It defines the ratio of the adjacent side to the opposite side in a right-angled triangle, making it critical for solving various trigonometric equations.
Understanding the properties of the cotangent function, such as its periodicity and symmetry, is vital for identifying its values for specific angles within defined intervals.

• Periodicity: The cotangent function is periodic with a period of \( \pi \), meaning \( \cot(x + \pi) = \cot x \) for any angle \( x \).
• Symmetry: It is symmetric with respect to the origin, which means it is an odd function: \( \cot(-x) = -\cot x \).

When solving an equation involving \( \cot x \), having insights into these characteristics allows for easier identification of angles where \( \cot x \) achieves particular values, such as those encountered in the step-by-step solution.

Solutions Over an Interval

When solving trigonometric equations, it is essential to find solutions within a specific interval—in this case, \( [0, 2\pi) \).
This interval specifies that solutions should be found between 0 and just before \( 2\pi \).
This range represents one full cycle of standard trigonometric functions. Within this interval, it is necessary to consider where each trigonometric function, like the cotangent, achieves the desired values.
Given their periodic nature, trigonometric functions may yield multiple solutions within any interval. Therefore, understanding the periodicity and behavior of the function is crucial to identifying all solutions.
Here’s a breakdown of finding solutions:

• First, determine the values for which the trigonometric function yields the necessary results.
• Next, apply the function’s period to enumerate all solutions within the interval.
• Finally, verify that all derived solutions fit within the defined interval's bounds.

Through this careful analysis, every potential solution within the interval is captured, ensuring comprehensive results for equations like those observed in trigonometric problems.