Solve 2cos²x = 3cosx - 1: Step-by-Step Guide

Hey guys! Today, we're diving into a trigonometric equation that might look a bit intimidating at first, but trust me, it's totally manageable once we break it down. We're going to solve the equation 2 cos² x = 3 cos x - 1. This is a classic example of a trigonometric equation that can be solved using algebraic techniques, specifically by transforming it into a quadratic equation. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We need to find all the values of x that satisfy the equation 2 cos² x = 3 cos x - 1. Remember, x represents an angle, and the cosine function gives us the x-coordinate of a point on the unit circle corresponding to that angle. So, we're essentially looking for angles where twice the square of the cosine is equal to three times the cosine minus one. This type of problem often appears in calculus, physics, and engineering, so mastering it is a fantastic way to boost your problem-solving skills.

Recognizing the Quadratic Form

The key to solving this equation is to recognize that it's quadratic in disguise. Notice how we have a term with cos² x, a term with cos x, and a constant term. This is the hallmark of a quadratic equation. To make this clearer, we can use a substitution. Let's say y = cos x. If we substitute y into our original equation, we get:

2y² = 3y - 1

Now, doesn't that look more familiar? This is a quadratic equation in the variable y. Our goal now is to solve for y, and then we can go back and find the values of x that correspond to those y values. This technique of using substitution to simplify equations is a powerful tool in mathematics, and you'll find it useful in many different contexts. For example, in physics, you might use a similar approach to solve differential equations, and in computer science, you might use substitution to simplify complex algorithms.

Why This Matters

Understanding how to solve trigonometric equations like this is super important for a bunch of reasons. First off, it's a fundamental skill in trigonometry itself. Trigonometric equations pop up all over the place, from calculating angles in triangles to modeling periodic phenomena like waves and oscillations. Secondly, this type of problem helps you develop your algebraic manipulation skills. You're not just memorizing formulas; you're learning how to rearrange equations, make substitutions, and apply different problem-solving strategies. This kind of thinking is crucial for success in higher-level math courses and in many STEM fields. Finally, tackling these problems builds your confidence. When you see a tricky equation and you know you have the tools to solve it, that's a great feeling!

Step-by-Step Solution

Alright, let's get down to the nitty-gritty and walk through the solution step-by-step. We'll take it nice and slow, so you can follow along easily. Remember, the goal is not just to get the answer, but to understand the process. That's what will really help you in the long run.

Step 1: Rearrange the Equation

Our first step is to rearrange the equation so that it's in standard quadratic form, which is ay² + by + c = 0. To do this, we need to move all the terms to one side of the equation. We have:

2y² = 3y - 1

Subtract 3y from both sides and add 1 to both sides:

2y² - 3y + 1 = 0

Now we have a quadratic equation in the familiar form. This is a crucial step because many techniques for solving quadratic equations, like factoring and the quadratic formula, rely on having the equation in this standard form. You can think of this step as setting the stage for the rest of the solution. If you don't get the equation into the correct form, the subsequent steps will be much more difficult, if not impossible.

Step 2: Solve the Quadratic Equation

Now that we have our quadratic equation in standard form, we can solve for y. There are a couple of ways to do this: factoring or using the quadratic formula. Let's try factoring first, as it's often the quickest method if it works. We're looking for two numbers that multiply to give (2)(1) = 2 and add up to -3. Those numbers are -2 and -1. So, we can rewrite the middle term as:

2y² - 2y - y + 1 = 0

Now, we can factor by grouping:

2y(y - 1) - 1(y - 1) = 0

(2y - 1)(y - 1) = 0

Setting each factor equal to zero gives us our solutions for y:

2y - 1 = 0 => y = 1/2 y - 1 = 0 => y = 1

So, we have two possible values for y: 1/2 and 1. Factoring is a powerful technique, but it doesn't always work. If you can't easily factor the quadratic, the quadratic formula is your trusty backup. Remember, the quadratic formula is:

y = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 2, b = -3, and c = 1. Plugging these values into the quadratic formula would also give us y = 1/2 and y = 1. The key is to choose the method you're most comfortable with and the one that seems most efficient for the particular problem. With practice, you'll get a feel for which method is best in different situations.

Step 3: Substitute Back and Solve for x

Remember that y was just a temporary placeholder for cos x. Now we need to substitute back to find the values of x. We have two equations to solve:

cos x = 1/2 cos x = 1

Let's tackle cos x = 1/2 first. We need to find the angles x whose cosine is 1/2. Think about the unit circle. Cosine corresponds to the x-coordinate, so we're looking for points on the unit circle where the x-coordinate is 1/2. These angles are x = π/3 and x = 5π/3 (or 60° and 300°) within the interval [0, 2π). But remember, the cosine function is periodic, meaning it repeats itself every 2π radians. So, we need to add multiples of 2π to these solutions to get all possible solutions:

x = π/3 + 2πk, where k is an integer x = 5π/3 + 2πk, where k is an integer

Now, let's consider cos x = 1. This occurs when x = 0 (or 0°). Again, due to the periodicity of the cosine function, we need to add multiples of 2π:

x = 2πk, where k is an integer

These are all the solutions to our original equation! Substituting back and considering the periodicity of trigonometric functions is a crucial step in solving trigonometric equations. If you forget this step, you'll only find a limited set of solutions, and you'll miss out on the infinitely many other solutions that exist due to the periodic nature of these functions.

General Solutions

So, to recap, the general solutions to the equation 2 cos² x = 3 cos x - 1 are:

• x = π/3 + 2πk
• x = 5π/3 + 2πk
• x = 2πk

where k is any integer. These solutions represent all the angles that satisfy the given equation. We use the term "general solutions" to emphasize that we're not just looking for solutions within a specific interval, like [0, 2π), but rather all possible solutions across the entire domain of the cosine function. Understanding general solutions is essential because it gives you a complete picture of the behavior of the trigonometric function and its solutions.

Tips and Tricks

Here are a few extra tips and tricks to keep in mind when solving trigonometric equations:

• Always check for the quadratic form: If you see a trigonometric function squared and the same function to the first power, think quadratic! Use substitution to make it clearer.
• Factor if possible: Factoring is usually the fastest way to solve a quadratic equation, but don't be afraid to use the quadratic formula if factoring seems difficult.
• Remember the unit circle: The unit circle is your best friend when solving trigonometric equations. It helps you visualize the angles and their corresponding sine and cosine values.
• Consider periodicity: Don't forget to add 2πk (or πk for tangent) to your solutions to account for the periodic nature of trigonometric functions.
• Check your solutions: Plug your solutions back into the original equation to make sure they work. This is a critical step to catch any errors you might have made along the way.

Practice Makes Perfect

The best way to get comfortable with solving trigonometric equations is to practice, practice, practice! Work through as many examples as you can, and don't be afraid to make mistakes. Mistakes are learning opportunities! The more you practice, the better you'll become at recognizing patterns, applying the correct techniques, and avoiding common pitfalls. You can find tons of practice problems online, in textbooks, and from your instructors. Don't just focus on getting the right answer; focus on understanding the process and the reasoning behind each step. That's what will truly make you a master of trigonometry.

Conclusion

So, there you have it! We've successfully solved the equation 2 cos² x = 3 cos x - 1. Remember the key steps: recognize the quadratic form, rearrange the equation, solve for the trigonometric function, substitute back, and account for periodicity. With a little practice, you'll be solving these types of equations like a pro. Keep up the great work, and happy solving!