Solving Cos²x - Cosx - 2 = 0: Factoring Trigonometric Equations In [0, 2π]

Hey guys! Let's dive into solving a trigonometric equation by factoring. This is a super useful skill, especially when you're dealing with trig functions within a specific interval. We're going to tackle the equation cos²x - cosx - 2 = 0 where x lies between 0 and 2π. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving this trigonometric equation, let's break down exactly what we're dealing with. Our mission, should we choose to accept it (and we do!), is to find all the values of x that make the equation cos²x - cosx - 2 = 0 true. The catch? These x values need to fall within the interval of 0 to 2π. This interval is super important because it represents one full revolution around the unit circle. Think of it as finding all the angles within one complete circle where the equation holds up.

Now, let's zoom in on the equation itself: cos²x - cosx - 2 = 0. Notice anything familiar? It might look a little intimidating at first glance, but it's actually a quadratic equation in disguise! We've got a squared term (cos²x), a linear term (-cosx), and a constant term (-2). This is a classic setup for factoring, which is the technique we'll be using to crack this problem. Factoring is a powerful tool that allows us to break down complex expressions into simpler parts, making it easier to find solutions. By recognizing this quadratic structure, we can apply familiar algebraic techniques to solve a trigonometric puzzle. It's like finding a hidden pattern that unlocks the solution!

The Factoring Process

Okay, guys, let's roll up our sleeves and dive into the heart of the matter: the factoring process. Remember how we spotted that our trigonometric equation, cos²x - cosx - 2 = 0, is secretly a quadratic equation? Well, we're going to use that to our advantage! To make things crystal clear, let's do a little substitution trick. We'll replace cosx with a simple variable, let's say y. This transforms our equation into something super familiar: y² - y - 2 = 0. See? Much less intimidating already!

Now, we need to factor this quadratic equation. Factoring is like finding the two numbers that multiply to give us the constant term (-2) and add up to give us the coefficient of the linear term (-1). In this case, those numbers are -2 and +1. Why? Because (-2) * (+1) = -2 and (-2) + (+1) = -1. So, we can rewrite our equation in factored form as: (y - 2)(y + 1) = 0. We're on the right track, guys!

But remember, we're not solving for y, we're solving for x! So, we need to undo our substitution. We'll replace y with cosx again, giving us: (cosx - 2)(cosx + 1) = 0. Now we've got our original equation factored, and we're ready to take the next step towards finding the solutions for x.

Solving for cosx

Alright, now that we've factored our equation into (cosx - 2)(cosx + 1) = 0, we're in a fantastic position to solve for x. The key here is to remember the zero product property. This super helpful rule states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, that means either (cosx - 2) = 0 or (cosx + 1) = 0. Let's tackle each of these possibilities separately.

First, let's consider cosx - 2 = 0. If we add 2 to both sides of the equation, we get cosx = 2. Now, think about the cosine function. Remember that the cosine of any angle always falls between -1 and 1, inclusive. It's a fundamental property of the cosine function. So, can cosx ever be equal to 2? Nope! This means that the equation cosx = 2 has no solutions. We can safely ignore this part and focus on the other factor.

Next up, we have cosx + 1 = 0. Subtracting 1 from both sides gives us cosx = -1. Ah, this looks much more promising! We know that cosine represents the x-coordinate on the unit circle. So, we're looking for the angles where the x-coordinate is -1. Can you picture it? There's only one point on the unit circle where this happens: at the angle π (or 180 degrees). So, we've found a potential solution: x = π. But hold on, we're not quite done yet! We need to make sure this solution fits within our given interval and if there are any other solutions within the interval [0, 2π].

Finding Solutions within the Interval [0, 2π]

Okay, we've made some serious progress! We know that cosx = -1 leads to a potential solution of x = π. But remember, our mission is to find all the solutions within the interval of 0 to 2π. So, we need to make sure our solution fits the bill and see if there are any other sneaky solutions hiding within that interval.

Let's start by checking if x = π is indeed a valid solution within our interval. Is π between 0 and 2π? Absolutely! π is approximately 3.14, which definitely falls within that range. So, we've confirmed that x = π is one of our solutions. Awesome!

Now, the crucial question: are there any other angles within the interval 0 to 2π that also satisfy cosx = -1? To answer this, let's think about the unit circle again. We're looking for points where the x-coordinate is -1. As we travel around the unit circle from 0 to 2π, we hit the point with an x-coordinate of -1 only once, and that's at π. There's no other place on the unit circle within that interval where the cosine is -1.

So, we've done it! We've thoroughly investigated the interval and found that x = π is the only solution to our equation cosx = -1 within the given range. This means we've cracked the code and found all the solutions to our original trigonometric equation.

Final Answer

Alright, guys, after a bit of trigonometric detective work, we've reached the finish line! We set out to solve the equation cos²x - cosx - 2 = 0 within the interval 0 to 2π, and we've successfully tracked down all the solutions. Through the power of factoring and a little help from the unit circle, we've discovered that there's only one value of x that makes this equation true within our specified range.

So, the final answer is: x = π. That's it! We've conquered this trigonometric challenge. Give yourselves a pat on the back – you've earned it!

Solve for x in the equation cos²x - cosx - 2 = 0, where 0 ≤ x ≤ 2π.

Solving cos²x - cosx - 2 = 0 Factoring Trigonometric Equations in [0, 2π]