Problem 19 - Thomas Calculus in SI Units

Question

At what value(s) of \(x\) does \(\cos x=2 x ?\)

Step-by-Step Solution

Answer

There are no values of \( x \) where \( \cos x = 2x \), within the constraints of the problem.

Step 1: Understanding the Problem

We have the equation \( \cos x = 2x \) and we need to find the value(s) of \( x \) for which this equation holds true. The function \( \cos x \) is a trigonometric function and \( 2x \) is a linear function, and we are looking for points where the graph of \( \cos x \) intersects the graph of \( 2x \).

Step 2: Analyzing the Range of Functions

The \( \cos x \) function has a range from -1 to 1 for all real numbers, while \( 2x \) is a linear function with a range of all real numbers. Since \( 2x \) can be greater than 1 or less than -1, \( \cos x = 2x \) can only have solutions where \( x \) keeps \( 2x \) within the range -1 to 1.

Step 3: Setting Constraints for \( x \)

Given \( 2x \) must be in the range \([-1, 1]\), we set up inequalities: 1. \( 2x \geq -1 \) which simplifies to \( x \geq -\frac{1}{2} \).2. \( 2x \leq 1 \) which simplifies to \( x \leq \frac{1}{2} \).Therefore, \( x \) must satisfy \(-\frac{1}{2} \leq x \leq \frac{1}{2}\).

Step 4: Finding Intersection Points

Within the range \(-\frac{1}{2} \leq x \leq \frac{1}{2}\), we check if there are intersections:- Check \( x = 0 \): \( \cos 0 = 1 \), and \( 2(0) = 0 \). These do not equal. Testing other values in the interval visually or using a calculator confirms that there is no \( x \) within this range where \( \cos x = 2x \) holds, or alternatively the graphs can be plotted to show that they don't intersect.

Key Concepts

Cosine Function

The cosine function is both fascinating and vital in trigonometry. It showcases how mathematical equations can model periodic behaviors or 'waves.' The function is denoted as \( \cos \theta \) where \( \theta \) generally represents an angle. Here's what you should know:

• Periodic Nature: Cosine repeats its values every \(2\pi\) radians. This means that \( \cos(\theta + 2\pi) = \cos(\theta) \).
• Range and Domain: The range of the cosine function is from -1 to 1, while the domain is all real numbers.
• Key Points: At \( \theta = 0 \), \( \cos 0 = 1 \). Then, as \( \theta \) increases to \( \pi/2 \) radians, \( \cos \theta \) decreases to 0.

Understanding these characteristics helps when solving equations that involve cosines, such as the equation \( \cos x = 2x \). Here, the range limitation of \( \cos x \) is particularly important.

Linear Equations

Linear equations describe straight-line graphs. They are among the most accessible types of equations and can provide foundational insights when advanced topics arise. The general form is \( ax + b = 0 \), where \( a \) and \( b \) are constants:

• Slope: The coefficient \( a \) in \( y = ax + b \) determines the slope of the line. In our exercise, the slope is 2, as seen in \( y = 2x \).
• Y-intercept: This is where the line crosses the y-axis. The term \( b \) determines this in the general form. If \( b = 0 \), the line passes through the origin.

These lines have no bounds, unlike trigonometric functions, and can extend infinitely in either direction. This characteristic becomes key in exercises contrasting such linear functions with the bounded range of cosine functions.

Intersection of Functions

When dealing with equations such as \( \cos x = 2x \), finding the intersection points is crucial as it provides the values solving the equation. Understanding this concept involves a few steps:

• Graphically: By plotting \( y = \cos x \) and \( y = 2x \), intersections are where the graphs meet. This visual method is helpful when analytical methods are complex or cumbersome.\li>Range Constraints: Since \( \cos x \) has a range of \(-1,1\), the possible \( x \) values for which \( 2x \) can equal \( \cos x \) is significantly narrowed down.
• Numeric Checks: After constraining possible \( x \) values, check them individually. However, as in this exercise, sometimes no real solutions satisfy both sides of the equation.

Thus, intersections not found directly become apparent when one function's limits prevent any shared solution with another.