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Chapter 4: Problem 64

Find all real numbers that satisfy each equation. Round approximate answers to the nearest hundredth. $$ 4-\cot (\pi x-2)=0 $$

Short Answer

Expert verified
The approximate solutions to the nearest hundredth are \( x \approx 0.72 \) and \( x \approx 1.72 \).

Step by step solution

01

Isolate the Cotangent Expression

Start by isolating the cotangent function. Subtract 4 from both sides of the equation: \[ 4 - \text{cot}(\pi x - 2) = 0 \]rearranges to \[ - \text{cot}(\pi x - 2) = -4 \]which simplifies to: \[ \text{cot}(\pi x - 2) = 4 \]
02

Take the Inverse Cotangent

To solve for \(\pi x - 2\), take the inverse cotangent of both sides: \[ \pi x - 2 = \text{cot}^{-1}(4) \]Let's denote this as \( \theta = \text{cot}^{-1}(4) \). Using a calculator or a table of trigonometric values, we find that: \[ \theta \approx 0.245 \]
03

Solve for \(x\)

Rearrange the earlier equation to solve for \(x\): \[ \pi x - 2 = 0.245 + k\pi \]for integer values of \(k\), since the cotangent function is periodic with period \(\pi\). Now solve for \(x\): \[ \pi x = 2 + 0.245 + k\pi \]\[ x = \frac{2 + 0.245 + k\pi}{\pi} \]
04

Simplify and Round the Result

Simplify \(x\) for different values of \(k\). For \(k = 0\): \[ x = \frac{2 + 0.245}{\pi} \approx 0.715 \] when rounded to the nearest hundredth. For \(k = 1\): \[ x = \frac{2 + 0.245 + \pi}{\pi} \approx 1.715 \] when rounded to the nearest hundredth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function
The cotangent function, denoted as \(\text{cot}(x)\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, so \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\). The cotangent function has some unique characteristics:
  • It has a periodicity of \(\pi\) rather than \(\frac{2\pi}{1}\).
  • It is undefined at the points where sine is zero, and this happens at integer multiples of \(\pi\).
  • The function is positive when the angle \(\theta\) is in the first and third quadrants, and negative in the second and fourth quadrants.
  • Graphically, it exhibits vertical asymptotes at integer multiples of \(\pi\).
Understanding the properties of the cotangent function is crucial when solving equations involving this trigonometric function. In this problem, we begin by isolating the cotangent term to solve for the unknown variable.
Inverse Cotangent
The inverse cotangent function, written as \(\text{cot}^{-1}(x)\), allows us to determine the angle whose cotangent is a given number. Mathematically, \(\text{cot}^{-1}(a)=\theta\) where \(\text{cot}(\theta)=a\). Here are some key points:
  • The range of \(\text{cot}^{-1}(x)\) is usually \((0, \pi)\), meaning the principal value lies between 0 and \(\pi\) radians.
  • Using a calculator or trigonometric table helps in finding the numerical value of \(\text{cot}^{-1}(x)\).
In our equation, \(\text{cot}(\pi x - 2) = 4\), the step of taking the inverse cotangent provides us with \(\theta \approx 0.245\). This value represents the principal solution, but remember multiple solutions exist because of the function's periodicity.
Trigonometric Periodicity
Trigonometric functions are periodic, meaning they repeat their values over specific intervals. The cotangent function has a period of \(\pi\), which greatly influences how we solve equations involving it. The periodicity tells us that if \(\text{cot}(\theta)=a\), then \(\text{cot}(\theta + k\pi)=a\) for any integer \(\text{k}\). This periodic nature allows us to find all possible solutions for \(\theta\). In our problem, we express the general solution as:
\(\pi x - 2 = 0.245 + k\bm{\pi}\). \ Here, \(\text{k}\) is an integer, covering all possible angles for which the cotangent is equal to 4.
Solving for x in Trigonometric Equations
To solve an equation involving trigonometric functions like cotangent, follow these systematic steps:
  • Isolate the trigonometric function (Here, that means isolating \(\text{cot}(\pi x - 2)\)).
  • Use the inverse trigonometric function to find a principal solution (Here, \(\text{cot}^{-1}(4)\rightarrow0.245\)).
  • Account for the periodicity to find the general solution (Add \(\bm{k\pi}\) in our step).
  • Solve for the variable concerned (Here, rearrange to solve for \(\text{x}\)).
    So we have \( x =\frac{2 + 0.245 + k \bm{\pi}}{\bm{\pi}}\).

By substituting different integer values of \(\text{k}\), we can find various solutions for \(\text{x}\). For example:
For \(\text{k=0}\): \( x \approx 0.715\), rounded to the nearest hundredth.
For \(\text{k=1}\): \( x \approx 1.715\), rounded to the nearest hundredth.
This ensures all potential solutions are identified based on the underlying periodic nature of the cotangent function.

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