Problem 21 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Understand the Problem

We are asked to find the value(s) of \(x\) where the function \( \cos x = 2x \). This means we need to find the intersection points between the cosine function and the line function \(y = 2x\).

Step 2: Analyze the Range of the Cosine Function

The cosine function, \( \cos x \), has a range from -1 to 1 for all real numbers \( x \). Therefore, for the equation \( \cos x = 2x \) to have solutions, it is necessary that \(-1 \leq 2x \leq 1\).

Step 3: Determine Possible Values for \( x \)

From the inequality \(-1 \leq 2x \leq 1\), we can solve for \( x \) by dividing each part of the inequality by 2. This gives us:\[-\frac{1}{2} \leq x \leq \frac{1}{2}\]. This suggests that any values for \(x\) that could potentially solve the equation must lie within this interval.

Step 4: Analyze Intersection Graphically or Numerically

The functions \(\cos x\) and \(2x\) can be plotted to visually inspect their intersection, or alternatively, a numerical method can be used to estimate where they intersect within the range \([-\frac{1}{2}, \frac{1}{2}]\).

Step 5: Solution by Trial or Numerical Method

Testing values within the range through numerical methods or by using solvers will show that the two functions intersect at \(x = 0\). Plugging \(0\) into the equation gives: \(\cos(0) = 2(0)\), verifying that \(1 = 0\), hence \(x = 0\) is the only solution.

Key Concepts

Cosine Function

The cosine function, symbolized as \( \cos x \), is a fundamental trigonometric function that depicts the horizontal coordinate of a point on the unit circle. The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. The importance of the cosine function lies in its periodic and oscillatory nature, making it vital in describing wave forms, cycles, and circular motion.

• **Range:** The cosine function oscillates between -1 and 1. This means that for any real number \( x \), the output \( \cos x \) is constrained within [-1, 1].
• **Periodicity:** With a period of \( 2\pi \), the cosine function repeats its values every \( 2\pi \) units.
• **Symmetry:** Cosine is an even function, meaning it is symmetrical around the y-axis, so \( \cos(-x) = \cos(x) \).

Recognizing the range is crucial when solving equations like \( \cos x = 2x \), because it helps determine the possible values \( x \) can take. Since \( 2x \) must also fit within this range, the equation must be analyzed over a domain where \(-1 \leq 2x \leq 1\). This restricts the potential solutions to a certain interval.

Numerical Methods

Numerical methods are mathematical techniques for approximating solutions to problems that might not be easy to solve analytically. These methods are especially useful in complex trigonometric equations, such as \( \cos x = 2x \), where finding an exact solution might not be straightforward. There are several types of numerical methods, including

• **Bisection Method:** A root-finding method that repeatedly bisects an interval and then selects a subinterval where a root must lie.
• **Newton's Method:** An iterative technique for finding successively better approximations to the roots (or zeroes) of a real-valued function.
• **Graphical Method:** By plotting the functions visually, one can estimate the point of intersection.

In the problem \( \cos x = 2x \), one approach is to graph the functionally oriented lines given by \(y = \cos x\) and \(y = 2x\). Visually inspecting these plots can help identify potential intersections, which can be further refined using numerical methods. In our specific case, "trial and error" within the restricted domain, \([-\frac{1}{2}, \frac{1}{2}]\), demonstrates that \(x = 0\) is the solution as it satisfies \(\cos(0) = 2(0)\).

Inequalities in Trigonometry

Inequalities are a potent tool in trigonometry and play a crucial role in determining the possible solutions to trigonometric equations. In the context of \( \cos x = 2x \), solving inequalities helps in establishing the domain within which we need to search for solutions.

The problem presents the range restriction \(-1 \leq 2x \leq 1\). Breaking it down gives us two simpler inequalities: \(-1 \leq 2x\) and \(2x \leq 1\). Solving these inequalities by dividing through by 2 results in \(x\) being bounded by \(-\frac{1}{2}\) and \(\frac{1}{2}\). This means:

• Any potential solution \(x\) must lie in the interval \([-\frac{1}{2}, \frac{1}{2}]\).
• The inequality ensures both functions are evaluated where \(2x\) remains a valid output for \(\cos x\) within its known range.

By restricting the search space, inequalities simplify complex equations by reducing the domain, thus facilitating the use of numerical methods or simple trials to find solutions.