Solving Cos(x+π) = 1/2 Trigonometric Equation On [π/2, Π] Interval
Trigonometric equations are a fundamental part of mathematics, appearing in various fields such as physics, engineering, and computer science. Mastering the techniques to solve these equations is crucial for anyone delving into these disciplines. This article will provide a comprehensive, step-by-step guide to solving the trigonometric equation $\cos (x+\pi)=\frac{1}{2}$ over the interval $\left[\frac{\pi}{2}, \pi\right]$. We'll break down each step, explaining the underlying concepts and reasoning, making it accessible for both students and professionals. Understanding these principles will empower you to tackle a wide range of trigonometric problems with confidence. Let's embark on this mathematical journey and unravel the solution together.
Understanding the Problem
Before diving into the solution, it's crucial to fully understand the problem. We are tasked with finding the values of x within the interval $\left[\frac\pi}{2}, \pi\right]$ that satisfy the equation $\cos (x+\pi)=\frac{1}{2}$. This involves several key concepts{2}, \pi\right]$, which represents a specific range of angles in radians, corresponding to the second quadrant of the unit circle. Fourth, understanding trigonometric identities, which allow us to manipulate and simplify equations. And fifth, being able to find general solutions and then narrow them down to the specific interval. Failing to understand any of these concepts can hinder finding the solution. By grasping these fundamental ideas, we can approach the problem methodically and arrive at the correct answer. Therefore, a solid grounding in these concepts is essential for successfully solving trigonometric equations. We will delve deeper into each of these aspects as we proceed through the solution process.
Step 1: Applying the Cosine Addition Formula
Our initial equation is $\cos (x+\pi)=\frac1}{2}$. To simplify this, we can leverage the cosine addition formula, a fundamental trigonometric identity. The cosine addition formula states that $\cos (A + B) = \cos A \cos B - \sin A \sin B$. Applying this to our equation, where A = x and B = , we get{2}$ now transforms into $- \cos x = \frac{1}{2}$. This is a significant simplification, as we have eliminated the compound angle (x + ) and are now dealing with a single cosine term. This step highlights the power of trigonometric identities in simplifying complex equations. By understanding and applying the appropriate identities, we can often reduce a problem to a more manageable form. This transformation makes it easier to isolate the variable and find the solutions. The application of the cosine addition formula is a crucial step in solving this equation, demonstrating a key technique in trigonometric problem-solving. This rewritten form allows us to proceed with finding the values of x that satisfy the equation.
Step 2: Solving for cos(x)
Following the simplification in the previous step, our equation is now $- \cos x = \frac1}{2}$. To isolate , we simply multiply both sides of the equation by -1. This gives us{2}$. This equation tells us that we are looking for angles x whose cosine value is -1/2. Recall that the cosine function represents the x-coordinate on the unit circle. Therefore, we are searching for points on the unit circle where the x-coordinate is -1/2. Now, we need to consider the interval given in the problem, which is $\left[\frac{\pi}{2}, \pi\right]$. This interval corresponds to the second quadrant of the unit circle. In the second quadrant, cosine values are negative, which aligns with our equation $\cos x = -\frac{1}{2}$. To find the angles x that satisfy this condition, we need to remember the special angles and their cosine values. We know that $\cos \frac{\pi}{3} = \frac{1}{2}$. Since we are looking for a negative cosine value in the second quadrant, we need to find the reference angle in the second quadrant that corresponds to $\frac{\pi}{3}$. This step of isolating is fundamental in solving trigonometric equations. It allows us to focus on the specific values of x that satisfy the given condition. The understanding of the unit circle and the signs of trigonometric functions in different quadrants is crucial at this point. We are now poised to identify the specific solutions within the given interval.
Step 3: Finding the Reference Angle and Solutions in the Interval
We've established that $\cos x = -\frac1}{2}$ and that we're looking for solutions in the interval $\left[\frac{\pi}{2}, \pi\right]$ (the second quadrant). To find the reference angle, we consider the angle whose cosine is $\frac{1}{2}$, without the negative sign. As mentioned earlier, we know that $\cos \frac{\pi}{3} = \frac{1}{2}$. Therefore, $\frac{\pi}{3}$ is our reference angle. In the second quadrant, the angle x that has a reference angle of $\frac{\pi}{3}$ is given by3}$. Calculating this, we get{3} - \frac{\pi}{3} = \frac{2\pi}{3}$. Now, we need to check if this solution lies within our given interval $\left[\frac{\pi}{2}, \pi\right]$. Since $\frac{\pi}{2} = \frac{3\pi}{6}$ and $\pi = \frac{6\pi}{6}$, we can see that $\frac{2\pi}{3} = \frac{4\pi}{6}$ falls between these values. Therefore, $\frac{2\pi}{3}$ is indeed a solution within the specified interval. It's important to note that trigonometric functions are periodic, meaning they repeat their values at regular intervals. However, because we are only considering a specific interval, we only need to find the solutions within that range. The process of finding the reference angle and then determining the angle in the correct quadrant is a key technique in solving trigonometric equations. This step demonstrates the importance of understanding the relationship between reference angles and angles in different quadrants. We have now successfully identified one solution to the equation within the given interval.
Step 4: Verifying the Solution
Having found a potential solution, x = $\frac2\pi}{3}$, it's crucial to verify that it indeed satisfies the original equation $\cos (x+\pi)=\frac{1}{2}$ and lies within the interval $\left[\frac{\pi}{2}, \pi\right]$. We've already confirmed that $\frac{2\pi}{3}$ is within the interval. Now, let's substitute this value into the original equation{3} + \pi) = \cos (\frac{2\pi}{3} + \frac{3\pi}{3}) = \cos (\frac{5\pi}{3})$. To evaluate $\cos (\frac{5\pi}{3})$, we can again use the unit circle or reference angles. The angle $\frac{5\pi}{3}$ lies in the fourth quadrant, where cosine values are positive. The reference angle for $\frac{5\pi}{3}$ is $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$. Therefore, $\cos (\frac{5\pi}{3}) = \cos (\frac{\pi}{3}) = \frac{1}{2}$. This confirms that x = $\frac{2\pi}{3}$ is indeed a solution to the original equation. Verification is a vital step in any mathematical problem-solving process. It ensures that the solution obtained is correct and that no errors were made in the intermediate steps. In trigonometric equations, verification is particularly important due to the periodic nature of the functions and the potential for extraneous solutions. By substituting the solution back into the original equation, we can be confident in our answer. This step highlights the importance of rigor and attention to detail in mathematics. Having verified our solution, we can confidently conclude that x = $\frac{2\pi}{3}$ is the correct solution within the given interval.
Conclusion: The Solution and Key Takeaways
In conclusion, the solution to the equation $\cos (x+\pi)=\frac{1}{2}$ over the interval $\left[\frac{\pi}{2}, \pi\right]$ is x = $\frac{2\pi}{3}$. This solution was obtained through a systematic approach involving several key steps. First, we applied the cosine addition formula to simplify the equation. Second, we solved for $\cos x$, which allowed us to isolate the trigonometric function. Third, we found the reference angle and identified the solution within the specified interval, leveraging our understanding of the unit circle and quadrantal angles. Fourth, we verified our solution by substituting it back into the original equation. This process highlights several important takeaways for solving trigonometric equations. First, understanding and applying trigonometric identities is crucial for simplifying complex equations. Second, a solid grasp of the unit circle and the behavior of trigonometric functions in different quadrants is essential for finding solutions within specific intervals. Third, verification is a vital step to ensure the accuracy of the solution. Fourth, a systematic and methodical approach is key to success in solving these types of problems. By mastering these techniques and concepts, you can confidently tackle a wide range of trigonometric equations. The ability to solve trigonometric equations is a valuable skill in various fields, and a thorough understanding of the underlying principles is the foundation for success. This step-by-step guide provides a solid framework for approaching these problems and arriving at the correct solutions. Remember to practice regularly and apply these techniques to different scenarios to further enhance your problem-solving abilities in trigonometry.
