91Ó°ÊÓ

Factoring is a technique used to simplify equations and make them easier to solve. In this problem, the equation is initially given as \(2 \theta \cos \theta + \theta = 0\). Notice that both terms in this equation have a common factor, \(\theta\). Factoring out \(\theta\) from the expression, we rewrite the equation as \(\theta(2 \cos \theta + 1) = 0\).

This transformation allows us to address each part separately. The factored form is crucial because it breaks down the equation into simpler components. Once factored, the properties of zero come into play, simplifying the solving process.

A general solution in trigonometric equations refers to all possible solutions which satisfy the equation. This is important because trigonometric functions are periodic, and hence have multiple solutions within any given interval.

For this problem, once the equation \(\theta(2 \cos \theta + 1) = 0\) was factored, it resulted in two separate equations to solve independently: \(\theta = 0\) and \(2 \cos \theta + 1 = 0\).

The solution \(\theta = 0\) is straightforward, while the equation \(2 \cos \theta + 1 = 0\) requires a few more steps. For it, the general solutions are described by the formulae \(\theta = \arccos\left(-\frac{1}{2}\right) + 2n\pi\) and \(\theta = -\arccos\left(-\frac{1}{2}\right) + 2n\pi\), where \(n\) is any integer.

This approach ensures that all of the infinite number of solutions are expressed.

The inverse cosine function, denoted as \(\arccos\), is a fundamental concept used to find angles when the cosine value is known. In this exercise, we ended up with the equation \(\cos \theta = -\frac{1}{2}\).

Using the \(\arccos\) function, we determine that \(\theta = \arccos\left(-\frac{1}{2}\right)\). The use of \(\arccos\) lets us retrieve the principal angle which has the given cosine value.

However, angles with the same cosine value recur in cycles. That's why the general solution also adds multiples of \(2\pi\), representing one full cycle of a cosine wave, to the principal angle, thereby capturing all possible solutions. This cyclic property underlies trigonometric functions' periodic nature.

A product is equal to zero when at least one of its factors equals zero. This is known as the Zero Product Property. In the factored equation \(\theta(2 \cos \theta + 1) = 0\), we use this property to divide the problem into manageable pieces.

This means either \(\theta = 0\) or \(2 \cos \theta + 1 = 0\). Breaking it down this way turns the problem into simpler parts that can each be solved on their own.

Recognizing such properties is extremely useful in algebra and trigonometry. It simplifies not only trigonometric equations but is also applicable in various algebraic situations, making problem-solving more efficient and less error-prone.