Solving 2cos²(θ) - 5cos(θ) = -2: A Step-by-Step Guide
Hey guys! Today, we're diving into a cool math problem: solving the trigonometric equation 2cos²(θ) - 5cos(θ) = -2. Don't worry, it might look a little intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. We'll be using some basic algebra and a bit of trig knowledge to get to the solution. The process involves treating the cosine function as a variable, which simplifies the equation into a more familiar quadratic form. Then, we will find the values of cos(θ) that satisfy the equation. Finally, we'll use our understanding of the unit circle or trigonometric functions to find the angles θ that correspond to those cosine values. This is a common type of problem you might encounter in a trigonometry class or even in calculus. So, let's get started and unravel this trigonometric puzzle together! Let's get our hands dirty and see how we can solve this problem. Ready? Let's go!
Step 1: Rearranging the Equation
Alright, the first thing we want to do is make this equation look a little friendlier. Our goal here is to get everything on one side of the equation and set it equal to zero. This is a classic move when dealing with quadratic equations, which, as you'll see, is exactly what we're going to create here. So, let's add 2 to both sides of the equation. This gives us:
2cos²(θ) - 5cos(θ) + 2 = 0
See? Doesn't that look better already? We've successfully transformed our equation into a standard quadratic form. This is crucial because it allows us to apply all the techniques we know for solving quadratic equations, like factoring or using the quadratic formula. The key is to recognize the structure and make the necessary adjustments to fit it into a familiar pattern. This initial step is often the most important because it sets the stage for the rest of the solution. If you set it up wrong, you will not get the correct answer. The other steps are just the process, but this is the foundation. Now that we have done the first step, let's move on to the next one.
Now, we are ready to proceed with the next step, where we can further simplify the equation and get closer to finding the solutions for θ. Make sure that you understand the process before going to the next step.
Step 2: Substitution (Making it Easier to Handle)
This is where things get really cool, guys. To make the equation easier to work with, we're going to use a little trick called substitution. We'll let x = cos(θ). This means, we're going to replace every instance of cos(θ) with x in our equation. That transforms our equation to:
2x² - 5x + 2 = 0
Boom! Doesn't that look much cleaner and more manageable? Now, we have a simple quadratic equation that we can solve using our favorite methods. Substitution is a super powerful tool in mathematics. It allows us to simplify complex expressions by replacing them with a single variable, making the problem easier to handle. In this case, it converts a trigonometric equation into a straightforward quadratic equation. This makes the solving process much more intuitive, because we're used to solving equations with variables like x and not with trigonometric functions. This trick is used all the time in all sorts of problems. It’s a great trick to have in your mathematical toolkit. Now, we're ready to find the roots of this equation, right? Let's move on!
This step is extremely useful, especially for beginners who are just starting to understand the concepts of trigonometry. The simplification makes it easier to focus on the algebraic aspects of the problem and reduces the cognitive load required to solve the equation. The transformation from a trigonometric equation to a standard quadratic equation simplifies the process and provides a clear pathway toward finding the solution.
Step 3: Factoring the Quadratic Equation
Okay, time to solve our quadratic equation, 2x² - 5x + 2 = 0. We're going to try to factor this bad boy. Factoring means we want to rewrite the equation as a product of two binomials. This is the goal, but sometimes it can be tricky. Here's how we can do it:
(2x - 1)(x - 2) = 0
To check if we factored correctly, you can multiply the binomials using the FOIL method (First, Outer, Inner, Last). If you do that, you'll see that it indeed gives you back the original quadratic equation. If you're not comfortable with factoring, you can always use the quadratic formula, but factoring is often quicker if you can spot it. So, we've successfully factored the equation. This is awesome because now we have a product of two terms that equals zero. This means that either the first term is zero, or the second term is zero (or both!). This is known as the zero-product property, and it's super important in algebra. If the product of two numbers is zero, then at least one of the numbers must be zero. This helps us find the possible values of x easily. Now, we will set each factor equal to zero and solve for x.
Factoring is a fundamental skill in algebra and is essential for solving quadratic equations. Mastering this skill significantly streamlines the solution process, making it faster and more efficient. The ability to factor quickly often distinguishes experienced mathematicians from those who are new to the subject.
Step 4: Solving for x
Alright, now we'll take each factor and set it equal to zero to solve for x. Remember that the factors we got were (2x - 1) and (x - 2). Let's start with the first one:
- 2x - 1 = 0
- Add 1 to both sides: 2x = 1
- Divide both sides by 2: x = 1/2
Great! We found our first solution. Now, let's do the second one:
- x - 2 = 0
- Add 2 to both sides: x = 2
Awesome, we have our two values of x. Remember, x = cos(θ), so we have found two possible values for cos(θ): 1/2 and 2. Remember, that means we need to find what angles give us these cosine values. Easy peasy!
Solving for x is a straightforward application of basic algebraic principles, such as isolating the variable by performing inverse operations. These steps are crucial because they directly lead to the possible values of x, which represent the cosine of the angles we are looking for. Careful execution of these steps ensures the accuracy of our solutions. The ability to solve for a variable is a fundamental skill that underpins many mathematical concepts. This skill is critical for finding the solutions to various equations.
Step 5: Finding the Values of θ
Now, this is where our knowledge of trigonometry kicks in! We have cos(θ) = 1/2 and cos(θ) = 2. But wait a minute, guys, we need to think carefully. The cosine function has a range of values between -1 and 1. This means, the value of cos(θ) can never be greater than 1 or less than -1. Therefore, cos(θ) = 2 is not a valid solution. So, we're going to ignore that one.
Now, let's focus on cos(θ) = 1/2. We need to find the angles θ for which the cosine is 1/2. We should recall the unit circle or the special triangles (like the 30-60-90 triangle) to help us. We know that cos(60°) = 1/2. But remember, the cosine function is positive in both the first and fourth quadrants. Therefore, another solution will be in the fourth quadrant. To find the angle in the fourth quadrant, we can use 360° - 60° = 300°.
So, the solutions for θ are:
- θ = 60° (or π/3 radians)
- θ = 300° (or 5π/3 radians)
Therefore, the solutions to the original equation 2cos²(θ) - 5cos(θ) = -2 are 60° and 300°. Remember, if the problem doesn't specify a range, the solutions can be generalized by adding multiples of 360° (or 2π radians) to the answers. And that's it! We solved it!
This final step brings everything together, using our trigonometric knowledge to find the actual angles that satisfy the equation. The ability to interpret the values of cosine and relate them to specific angles is a crucial skill in trigonometry. The use of the unit circle or special triangles is instrumental in finding the correct solutions. Understanding the limitations of the cosine function (its range between -1 and 1) is also important to identify valid solutions and avoid errors.
Step 6: Verification
It’s always a good idea to check your solutions. We can plug our found values back into the original equation to verify. For θ = 60°:
2cos²(60°) - 5cos(60°) = 2(1/2)² - 5(1/2) = 2(1/4) - 5/2 = 1/2 - 5/2 = -4/2 = -2
For θ = 300°:
2cos²(300°) - 5cos(300°) = 2(1/2)² - 5(1/2) = 2(1/4) - 5/2 = 1/2 - 5/2 = -4/2 = -2
Both values satisfy the equation, which confirms our solution!
This verification step is essential for ensuring the accuracy of our solutions. By substituting the found values of θ back into the original equation, we can confirm that they satisfy the equation. This not only validates our work but also reinforces our understanding of the trigonometric functions. The verification step is a critical part of problem-solving. This step ensures that any errors made during the solving process are caught early. This process also provides an opportunity to refine and improve the solution strategy. This step should never be skipped.
Conclusion
And there you have it, guys! We successfully solved the trigonometric equation 2cos²(θ) - 5cos(θ) = -2. We went through the steps of rearranging, substituting, factoring, solving, and then finding the angles. Remember that with practice, these types of problems will become easier and easier. The most important thing is to understand the underlying principles and to practice solving various problems. Keep at it, and you'll be a trig whiz in no time! Remember, math is all about practice and understanding. If you're struggling, don't worry! Go back, review the steps, and try again. And feel free to ask me any questions you may have. Good luck with your studies, and I hope this helped. Keep practicing, and you'll master these concepts!
In this guide, we've broken down a trigonometric equation into manageable steps, highlighting the importance of algebraic manipulation, substitution, and trigonometric identities. The successful solution demonstrates the synergy between algebra and trigonometry, and it underscores the importance of a systematic approach to problem-solving. It also highlights the significance of understanding fundamental concepts, such as the range of the cosine function and the properties of quadratic equations. By mastering these concepts, you can tackle more complex problems with confidence.
So, keep practicing, and never stop learning! I hope this was helpful! Bye!
