Solving 3 Csc X - 1 = 2 On The Interval [0, 2π] A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of trigonometry to solve a classic equation. We'll tackle the problem: 3 csc x - 1 = 2 within the interval [0, 2π). Don't worry if it looks intimidating – we'll break it down step-by-step. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what we're dealing with. The equation 3 csc x - 1 = 2 involves the cosecant function, which is a reciprocal trigonometric function. Remember, csc x is the same as 1/sin x. The interval [0, 2π) represents all angles from 0 to 2π radians, which is a full circle on the unit circle. Our goal is to find all the angles within this interval that satisfy the given equation. This involves manipulating the equation to isolate the trigonometric function, finding reference angles, and then determining the solutions within the specified interval. Each step in this process is crucial to ensure we arrive at the correct solutions and fully grasp the concepts involved.

The first key step in solving trigonometric equations like 3 csc x - 1 = 2 is to isolate the trigonometric function. In this case, we need to get csc x by itself on one side of the equation. To do this, we'll start by adding 1 to both sides of the equation. This gives us 3 csc x = 3. Next, we'll divide both sides by 3 to completely isolate csc x, resulting in csc x = 1. This simplified equation is much easier to work with. Once the trigonometric function is isolated, we can then use our knowledge of trigonometric identities and the unit circle to find the values of x that satisfy the equation. This initial isolation step is fundamental to solving a wide range of trigonometric problems, making it an essential technique to master.

Now that we have csc x = 1, the next step is to relate this back to the sine function. Since csc x is the reciprocal of sin x, we can rewrite the equation as 1/sin x = 1. This is equivalent to sin x = 1. Thinking in terms of sine makes it easier to visualize on the unit circle. We're looking for angles where the y-coordinate (which represents the sine value) is equal to 1. This translation from cosecant to sine is a critical step because sine is a more commonly used and easily visualized trigonometric function. Understanding this reciprocal relationship helps in solving a broader range of trigonometric equations. By converting to sine, we can more easily identify the angles on the unit circle that satisfy the original equation, leading us closer to the final solution.

Solving the Equation

Now that we've simplified the equation to sin x = 1, we need to find the angles x in the interval [0, 2π) where the sine function equals 1. Remember, on the unit circle, the sine value corresponds to the y-coordinate of a point on the circle. So, we're looking for points where the y-coordinate is 1. This occurs at only one point on the unit circle within the interval [0, 2π), which is at the angle π/2 (90 degrees). At this angle, the coordinates are (0, 1), confirming that sin(π/2) = 1. There are no other angles within the specified interval where the sine function equals 1. Thus, the solution to our equation is x = π/2. Visualizing the unit circle and understanding the sine function's behavior is essential for solving trigonometric equations efficiently and accurately. This step highlights the importance of connecting algebraic manipulations with geometric interpretations in trigonometry.

Detailed Solution Steps

Let's break down the solution into clear, easy-to-follow steps:

  1. Isolate the cosecant function:

    • Start with the equation: 3 csc x - 1 = 2
    • Add 1 to both sides: 3 csc x = 3
    • Divide by 3: csc x = 1

  2. Rewrite in terms of sine:

    • Since csc x = 1/sin x, we have 1/sin x = 1
    • This simplifies to sin x = 1

  3. Find the angles where sin x = 1:

    • On the unit circle, sin x = 1 at x = π/2

  4. Check the interval:

    • π/2 is within the interval [0, 2π)

Therefore, the solution to the equation 3 csc x - 1 = 2 on the interval [0, 2π) is x = π/2. Each of these steps is vital in reaching the correct solution. Isolating the trigonometric function is crucial for simplification, while converting to sine helps in visualizing the problem on the unit circle. Identifying the correct angles and ensuring they fall within the given interval are the final steps in confirming the solution's validity. Understanding and practicing these steps will significantly improve your ability to solve trigonometric equations.

Checking the Answer

It's always a good idea to check our solution to make sure it's correct. We found that x = π/2. Let's plug this value back into the original equation:

3 csc(π/2) - 1 = 2

Since csc(π/2) = 1/sin(π/2) = 1/1 = 1, the equation becomes:

3(1) - 1 = 2

3 - 1 = 2

2 = 2

The equation holds true, so our solution x = π/2 is correct! This verification step is crucial in mathematical problem-solving, as it ensures that the solution obtained satisfies the original equation. By substituting the solution back into the original equation, we can confirm its accuracy and avoid potential errors. This practice reinforces the understanding of the problem and the correctness of the solution process. Always take the time to check your answers, especially in exams or when dealing with complex equations.

Common Mistakes to Avoid

When solving trigonometric equations, there are a few common mistakes that students often make. Let's highlight some of them so you can avoid these pitfalls:

  1. Forgetting the reciprocal relationship: One common mistake is forgetting that csc x = 1/sin x. Always remember this fundamental relationship to correctly rewrite and solve equations. Keeping trigonometric identities like this one fresh in your mind is crucial for success. Trigonometry relies heavily on these relationships, and overlooking them can lead to incorrect solutions.

  2. Incorrectly isolating the trigonometric function: Make sure you perform the algebraic manipulations correctly to isolate the trigonometric function. For example, adding or dividing incorrectly can lead to a wrong answer. Double-checking your steps and ensuring you're following the correct order of operations is essential. Simple arithmetic errors can sometimes derail the entire solution process.

  3. Missing solutions within the interval: Remember to consider all possible solutions within the given interval. The unit circle is your best friend here! Visualize the angles and their corresponding sine, cosine, and tangent values to ensure you haven't missed any solutions. Trigonometric functions are periodic, meaning they repeat their values, so there may be multiple solutions within a given interval.

  4. Not checking the answer: Always check your solution by plugging it back into the original equation. This will help you catch any errors you might have made along the way. Checking your work is a fundamental part of problem-solving, not just in trigonometry but across all areas of mathematics. It provides confidence in your answer and helps reinforce the concepts you've used.

By being aware of these common mistakes, you can approach trigonometric equations with greater confidence and accuracy. Practice and careful attention to detail are key to mastering these types of problems.

Alternative Solution Methods

While we've covered the primary method for solving the equation 3 csc x - 1 = 2, it's always beneficial to explore alternative approaches. Understanding different methods can deepen your grasp of the concepts and provide you with more tools to tackle complex problems. While the core steps remain the same, slight variations in the order or approach can sometimes offer a clearer path to the solution. By expanding your problem-solving toolkit, you'll be better equipped to handle a variety of trigonometric equations and challenges.

  1. Graphical Approach: Another way to solve this equation is by using a graphical approach. We can graph the function y = 3 csc x - 1 and the line y = 2 on the same coordinate plane within the interval [0, 2π). The solutions to the equation are the x-coordinates of the points where the graph of the function intersects the line. This method provides a visual representation of the solutions and can be particularly helpful for understanding the behavior of trigonometric functions. The graphical approach can also be useful for estimating solutions when algebraic methods are more complex or time-consuming. While it may not always provide an exact answer, it offers valuable insight into the problem's nature.

  2. Using Trigonometric Identities Directly: Although we converted to sine, one could also think about the properties of the cosecant function directly on the unit circle. Remember that csc x is the reciprocal of sin x, so csc x = 1 means that sin x = 1. The angle at which the sine function equals 1 within the given interval is π/2. This direct approach leverages your understanding of trigonometric identities and the unit circle, allowing for a more intuitive solution process. It's a testament to the importance of mastering these fundamentals in trigonometry. By thinking about the properties of cosecant, we can bypass the step of converting to sine and arrive directly at the solution, showcasing the flexibility and interconnectedness of trigonometric concepts.

Conclusion

Great job, guys! We've successfully solved the equation 3 csc x - 1 = 2 on the interval [0, 2π). We found that the solution is x = π/2. Remember the key steps: isolate the trigonometric function, rewrite in terms of sine, find the angles on the unit circle, and check your answer. Keep practicing, and you'll become a trigonometry master in no time! Understanding these steps is vital for tackling more complex trigonometric problems. Practice not only reinforces the techniques but also deepens your understanding of the underlying concepts. As you work through more equations, you'll develop a stronger intuition for trigonometric functions and their relationships. This problem-solving experience is invaluable in solidifying your knowledge and building confidence in your abilities. So, keep at it, and you'll find that even the most challenging trigonometric equations become manageable with the right approach and consistent practice.