Solving Tan(arccos(-√2/2)) A Step-by-Step Trigonometric Exploration

In the realm of trigonometry, evaluating expressions involving inverse trigonometric functions can often seem like navigating a complex maze. However, with a systematic approach and a firm grasp of fundamental principles, these problems can be demystified. This article delves into the evaluation of $\tan(\arccos(-\frac{\sqrt{2}}{2}))$, offering a comprehensive, step-by-step solution and emphasizing the underlying concepts. Our main objective is to break down this problem into manageable steps, ensuring a clear and thorough understanding for anyone venturing into the world of trigonometry. This exploration will not only enhance your problem-solving skills but also deepen your appreciation for the elegance and interconnectedness of trigonometric functions. By the end of this article, you will be equipped to tackle similar challenges with confidence and precision.

At the heart of our problem lies the inverse cosine function, denoted as ${\arccos(x)}$ or ${\cos^{-1}(x)}$. It's crucial to grasp what this function represents: it essentially answers the question, "What angle has a cosine of ${x}$?" The range of ${\arccos(x)}$ is restricted to ${[0, \pi]}$, meaning the output angle will always fall within this interval. This restriction is essential to ensure that the inverse cosine function is well-defined, as the cosine function itself is periodic and would otherwise have infinitely many angles corresponding to a single cosine value. When we encounter ${\arccos(-\frac{\sqrt{2}}{2})}$, we're seeking an angle within the range of ${[0, \pi]}$ whose cosine is ${-\frac{\sqrt{2}}{2}}$. Recognizing this fundamental concept is the first step in unlocking the solution to our trigonometric puzzle. The interplay between inverse trigonometric functions and their corresponding trigonometric functions is a cornerstone of trigonometric problem-solving, and a solid understanding of this relationship is invaluable.

The initial step in solving ${\tan(\arccos(-\frac{\sqrt{2}}{2}))}$ is to determine the value of ${\arccos(-\frac{\sqrt{2}}{2})}$. We need to find an angle, let's call it ${\theta}$, within the range of ${[0, \pi]}$, such that ${\cos(\theta) = -\frac{\sqrt{2}}{2}}$. Recalling the unit circle and the values of cosine for standard angles, we know that ${\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}}$. However, we're looking for a negative value. Since cosine is negative in the second and third quadrants, we need to find an angle in the interval ${[0, \pi]}$ (which includes the first and second quadrants) where cosine is ${-\frac{\sqrt{2}}{2}}$. This angle is ${\frac{3\pi}{4}}$, as it lies in the second quadrant and its cosine is indeed ${-\frac{\sqrt{2}}{2}}$. Therefore, ${\arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}}$. This step highlights the importance of understanding the unit circle and the signs of trigonometric functions in different quadrants. Being able to quickly recall these values and relationships is crucial for efficiently solving trigonometric problems.

Now that we've established that ${\arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}}$, our problem simplifies to finding ${\tan(\frac{3\pi}{4})}$. The tangent function is defined as the ratio of sine to cosine, i.e., ${\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}}$. To find ${\tan(\frac{3\pi}{4})}$, we need to determine the sine and cosine of ${\frac{3\pi}{4}}$. As we previously discussed, ${\frac{3\pi}{4}}$ is in the second quadrant. In the second quadrant, sine is positive, and cosine is negative. We know that ${\sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}}$ and ${\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}}$. Therefore, ${\tan(\frac{3\pi}{4}) = \frac{\sin(\frac{3\pi}{4})}{\cos(\frac{3\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1}$. This calculation demonstrates how the values of sine and cosine in different quadrants directly influence the value of the tangent function. Understanding these relationships is essential for mastering trigonometric evaluations.

Putting it all together, we've successfully navigated the trigonometric expression ${\tan(\arccos(-\frac{\sqrt{2}}{2}))}$. We first determined that ${\arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}}$ and then calculated that ${\tan(\frac{3\pi}{4}) = -1}$. Therefore, the final answer is ${-1}$. This journey through the problem highlights the importance of breaking down complex expressions into smaller, more manageable steps. By systematically addressing each component, we were able to arrive at the solution with clarity and precision. This approach is applicable to a wide range of trigonometric problems, emphasizing the value of a methodical and conceptually sound problem-solving strategy. Our step-by-step solution underscores the significance of understanding inverse trigonometric functions, the unit circle, and the relationships between sine, cosine, and tangent in different quadrants. Each step built upon the previous one, illustrating the interconnectedness of trigonometric concepts and the power of a systematic approach.

In conclusion, the value of ${\tan(\arccos(-\frac{\sqrt{2}}{2}))}$ is -1. This problem serves as a valuable exercise in understanding and applying fundamental trigonometric principles. By systematically evaluating the inverse cosine function and then the tangent function, we arrived at the solution. The key takeaways from this exercise include the importance of understanding the range of inverse trigonometric functions, the unit circle, and the relationships between trigonometric functions in different quadrants. Mastering these concepts is essential for tackling more complex trigonometric problems. Furthermore, this problem underscores the value of a methodical approach to problem-solving, breaking down complex expressions into smaller, manageable steps. This strategy is not only effective in trigonometry but also applicable across various mathematical disciplines. We hope this detailed explanation has provided you with a clearer understanding of how to evaluate such expressions and has reinforced your trigonometric skills. Remember, practice is key to mastering these concepts, so continue to explore and challenge yourself with similar problems. The journey through trigonometry can be both challenging and rewarding, and with a solid foundation and a systematic approach, you can confidently navigate its intricacies.