Solving Cos(2x) - Sin(x) = 1 Trigonometric Equation In [0, 2π)

by Pedro Alvarez 63 views

Hey everyone! Let's dive into the fascinating world of trigonometry and tackle a problem that involves finding solutions to a trigonometric equation. Specifically, we're going to work through the equation cos(2x) - sin(x) = 1 and pinpoint all the solutions that fall within the interval of [0, 2π). This means we are looking for all the angles x between 0 and 2π (excluding 2π itself) that satisfy our equation. Trigonometric equations can seem a bit daunting at first, but with a strategic approach and some key trigonometric identities, we can break them down and find the answers we need. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into solving, let’s make sure we fully understand what the problem is asking. We have the trigonometric equation cos(2x) - sin(x) = 1, and our mission is to find all the values of x within the interval [0, 2π) that make this equation true. Remember, x represents an angle in radians. The interval [0, 2π) covers one full rotation around the unit circle, starting from 0 radians and going all the way up to (but not including) 2π radians. Think of it like finding the specific spots on a circle where the relationship between the cosine of twice the angle and the sine of the angle results in the equation equaling 1. This involves using our knowledge of trigonometric identities and algebraic manipulation to isolate x and identify those angles. Understanding this foundation is crucial because it guides our approach to solving the equation and interpreting the results. By breaking down the problem into its core components, we set ourselves up for a clearer and more effective solution process. Now that we have a solid grasp of the task at hand, we can move on to exploring the trigonometric identities and strategies that will help us find the solutions. Let's unravel this trigonometric puzzle together!

Key Trigonometric Identities

To effectively solve the equation cos(2x) - sin(x) = 1, we need to leverage some fundamental trigonometric identities. These identities are like the secret weapons in our mathematical arsenal, allowing us to rewrite and simplify equations into more manageable forms. One of the most crucial identities for this particular problem is the double-angle identity for cosine: cos(2x) = 1 - 2sin²(x). This identity is a game-changer because it allows us to express cos(2x) in terms of sin(x), which is the other trigonometric function present in our equation. By making this substitution, we can transform our original equation into one that involves only the sine function, making it significantly easier to solve. Think of it as converting the equation into a common language, allowing us to combine like terms and isolate the variable we're interested in. Furthermore, understanding the unit circle and the definitions of sine and cosine in terms of the unit circle is essential. The unit circle provides a visual representation of the values of sine and cosine for different angles, which helps us identify potential solutions and understand the periodic nature of these functions. Recognizing that sine and cosine repeat their values every 2π radians is key to finding all solutions within our specified interval [0, 2π). With these identities and concepts in mind, we're well-equipped to manipulate the equation and start our journey toward finding the specific values of x that satisfy it. Let’s dive deeper into how we can apply these tools to solve the problem step by step!

Solving the Equation Step-by-Step

Alright, let’s get our hands dirty and solve the equation cos(2x) - sin(x) = 1! Here’s a step-by-step breakdown to make things super clear.

  1. Substitute the Double-Angle Identity: Remember that crucial identity we talked about? We'll use cos(2x) = 1 - 2sin²(x) to replace cos(2x) in the original equation. This gives us: 1 - 2sin²(x) - sin(x) = 1. This substitution is a pivotal move because it unifies the trigonometric functions in the equation, making it much easier to handle. It's like translating different languages into one so that everyone can understand each other. By expressing everything in terms of sin(x), we’ve set the stage for algebraic manipulation and simplification.

  2. Simplify the Equation: Now, let's simplify things. Subtract 1 from both sides of the equation, and we get: -2sin²(x) - sin(x) = 0. This step is all about cleaning up the equation and making it more streamlined. By getting rid of the constant term, we pave the way for factoring and isolating the variable. Think of it as decluttering your workspace before starting a project – it makes the whole process smoother and more efficient.

  3. Factor out sin(x): Look closely! We can factor out sin(x) from the left side of the equation: sin(x)(-2sin(x) - 1) = 0. Factoring is a powerful technique in algebra, and it’s our key to unlocking the solutions here. By factoring out sin(x), we’ve transformed the equation into a product of two factors that equals zero. This is fantastic news because it means that either one or both of these factors must be zero, giving us two separate equations to solve. It’s like splitting a complex problem into smaller, more manageable pieces.

  4. Set Each Factor to Zero: Now, we'll use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations:

    • sin(x) = 0
    • -2sin(x) - 1 = 0

  5. Solve sin(x) = 0: Think about the unit circle! Where is the sine function equal to zero? Sine represents the y-coordinate on the unit circle, so sin(x) = 0 when x = 0, π. Visualizing the unit circle is incredibly helpful here. Remember, the sine function corresponds to the y-coordinate of a point on the unit circle. So, we're looking for the points where the y-coordinate is zero. These points occur at the angles 0 and π radians, which correspond to the positive x-axis and the negative x-axis, respectively. These are our first two solutions, and they're crucial stepping stones in finding all the solutions within the given interval.

  6. Solve -2sin(x) - 1 = 0: Let’s isolate sin(x). Add 1 to both sides and then divide by -2 to get: sin(x) = -1/2. Now, we need to find the angles where the sine function is equal to -1/2. This requires a bit more thought, but we're up for the challenge! We're essentially looking for the points on the unit circle where the y-coordinate is -1/2. This involves recalling our knowledge of special triangles and the symmetry of the unit circle.

  7. Find Angles for sin(x) = -1/2: Again, visualize the unit circle. Sine is negative in the third and fourth quadrants. The reference angle for sin(x) = 1/2 is π/6. So, the angles in the third and fourth quadrants where sin(x) = -1/2 are x = π + π/6 = 7π/6 and x = 2π - π/6 = 11π/6. The unit circle is our best friend here! We know that sine is negative in the third and fourth quadrants, so we need to find the angles in those quadrants that have a sine value of -1/2. By recalling our knowledge of special triangles (specifically the 30-60-90 triangle), we know that sin(π/6) = 1/2. The angle π/6 is our reference angle. To find the angles in the third and fourth quadrants, we add π/6 to π and subtract π/6 from 2π, respectively. This gives us the solutions x = 7π/6 and x = 11π/6.

  8. Check the Interval: Make sure all our solutions are within the interval [0, 2π). Lucky for us, all the solutions we found (0, π, 7π/6, 11π/6) fall within this interval. This is a crucial step in solving any trigonometric equation. We need to ensure that all the solutions we've found are actually within the specified interval. In our case, the interval is [0, 2π), which represents one full rotation around the unit circle. By checking that our solutions (0, π, 7π/6, and 11π/6) all fall within this interval, we confirm that they are valid solutions to the equation. This step is like the final quality check, ensuring that our answers are accurate and complete.

  9. List the Solutions: Finally, we can confidently state our solutions: x = 0, π, 7π/6, 11π/6. Congratulations! We've successfully navigated the twists and turns of this trigonometric equation and found all the solutions within the given interval. It's a testament to our understanding of trigonometric identities, algebraic techniques, and the unit circle. But, more importantly, it showcases our ability to break down a complex problem into manageable steps and persevere until we reach the final answer. Now, let's take a moment to reflect on the process and the key takeaways from this problem.

Final Solutions

So, after all that awesome work, we've pinpointed the solutions to the equation cos(2x) - sin(x) = 1 within the interval [0, 2π). Drumroll, please! The solutions are:

  • x = 0
  • x = π
  • x = 7π/6
  • x = 11π/6

These are the four angles within one full rotation of the unit circle that make our equation true. Each of these angles represents a point where the relationship between the cosine of twice the angle and the sine of the angle perfectly balances out to equal 1. It's like finding the specific coordinates on a map that lead us to our treasure. We've not only found the solutions but also reinforced our understanding of trigonometric functions, identities, and the unit circle. This knowledge will serve us well in tackling future trigonometric challenges. The process of solving this equation has been a journey of applying mathematical concepts, problem-solving strategies, and a bit of persistence. And now, we can confidently add these solutions to our collection of mathematical triumphs. But our journey doesn't end here. Let's take a moment to recap the key steps and insights we've gained from this problem, solidifying our understanding and preparing us for the next adventure in the world of mathematics!

Visualizing the Solutions

To really solidify our understanding, let's take a moment to visualize these solutions on the unit circle. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. The angles we found—0, π, 7π/6, and 11π/6—correspond to specific points on this circle.

  • x = 0: This is the point on the positive x-axis, where the circle intersects the axis. At this point, the cosine is 1 and the sine is 0.
  • x = π: This is the point on the negative x-axis, directly opposite x = 0. Here, the cosine is -1 and the sine is 0.
  • x = 7π/6: This angle is in the third quadrant, past π but not quite to 3π/2. In this quadrant, both cosine and sine are negative. This corresponds to a point in the third quadrant.
  • x = 11π/6: This angle is in the fourth quadrant, close to completing the full circle at 2π. Here, the cosine is positive, and the sine is negative. This corresponds to a point in the fourth quadrant.

Visualizing these solutions on the unit circle not only confirms that they fall within our interval of [0, 2π) but also gives us a deeper understanding of the behavior of trigonometric functions. We can see how the sine and cosine values change as we move around the circle, and how these values relate to the solutions of our equation. It's like seeing the equation come to life, transforming abstract symbols into concrete points on a circle. This visual representation is a powerful tool for reinforcing our understanding and making connections between different mathematical concepts. And now, with this visual understanding in place, let's wrap things up with a final recap of our journey and the key lessons we've learned.

Key Takeaways

Wow, guys, we really went through it all! Solving cos(2x) - sin(x) = 1 in the interval [0, 2π) wasn't just about finding the right numbers; it was a journey through trigonometric identities, algebraic manipulation, and the power of visualization. So, what did we learn along the way?

  • Trigonometric Identities are Your Friends: The double-angle identity for cosine, cos(2x) = 1 - 2sin²(x), was our MVP. Knowing your identities is like having a Swiss Army knife for trig problems! These identities are the fundamental building blocks of trigonometry, allowing us to rewrite and simplify complex expressions. They're like the secret codes that unlock the solutions to trigonometric equations. By mastering these identities, we gain the ability to transform equations into more manageable forms and bridge the gap between different trigonometric functions.

  • Factoring is a Powerful Tool: Factoring allowed us to break down a complex equation into simpler ones. Remember, if the product of factors is zero, at least one of them must be zero! This technique is a cornerstone of algebra and is essential for solving a wide range of mathematical problems. It's like taking a puzzle apart and examining each piece individually. By factoring, we can isolate the variables we're interested in and find the values that make the equation true.

  • The Unit Circle is Your Guide: The unit circle isn't just a pretty picture; it's a map to understanding sine, cosine, and their values at different angles. Visualizing solutions on the unit circle makes everything click. It's like having a visual representation of the entire trigonometric landscape. The unit circle allows us to connect angles to their corresponding sine and cosine values, visualize the periodicity of trigonometric functions, and identify solutions within a given interval. It's a powerful tool for both solving equations and developing a deeper understanding of trigonometry.

  • Step-by-Step Approach is Key: Breaking the problem into smaller, manageable steps made it way less intimidating. Don't try to eat the whole elephant at once! Just one bite at a time. This approach is a valuable strategy for tackling any complex problem, not just in mathematics. By breaking down a problem into smaller steps, we can focus on each step individually, making the overall process less overwhelming. It's like building a house brick by brick, rather than trying to construct the entire structure at once.

  • Always Check Your Interval: Never forget to check that your solutions fall within the specified interval. It's the final seal of approval on your hard work! This step is a crucial quality check that ensures our solutions are valid within the given context. By verifying that our solutions fall within the specified interval, we avoid including extraneous solutions and ensure that our answers are accurate and complete.

So, there you have it! We not only solved a challenging trigonometric equation but also reinforced some fundamental mathematical principles. Keep these takeaways in mind, and you'll be a trigonometric equation-solving rockstar in no time! Remember, math isn't just about getting the right answer; it's about the journey of discovery and the skills we develop along the way. And now, armed with our newfound knowledge and experience, we're ready to tackle the next mathematical challenge that comes our way. Let's keep exploring the fascinating world of mathematics together!