Arccos(-√3/2) Explained A Step-by-Step Solution

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The inverse cosine function, denoted as arccos(x) or cos⁻¹(x), plays a crucial role in trigonometry and various fields of mathematics and physics. Understanding the behavior and properties of this function is essential for solving trigonometric equations and dealing with angular relationships. In this comprehensive exploration, we will delve into the intricacies of the arccos function, specifically focusing on evaluating arccos(-√3/2). Our discussion will encompass the definition of arccos, its range and domain, the unit circle connection, and a step-by-step approach to finding the solution. Furthermore, we will analyze why the correct answer is 5π/6 and eliminate the other options, solidifying your understanding of inverse trigonometric functions.

To truly grasp the concept, let's begin by defining the inverse cosine function. The arccos(x) function answers the question: "What angle has a cosine of x?" In mathematical terms, if cos(θ) = x, then arccos(x) = θ. However, it's crucial to remember that the cosine function is periodic, meaning it repeats its values at regular intervals. To make the inverse cosine function well-defined, we restrict its range to [0, π]. This means that the output of arccos(x) will always be an angle between 0 and π radians (inclusive). The domain of arccos(x) is [-1, 1], as the cosine function's output always falls within this range. Grasping these domain and range restrictions is paramount to correctly evaluating arccos and avoiding potential pitfalls.

The unit circle provides a powerful visual aid for understanding trigonometric functions and their inverses. Imagine a circle with a radius of 1 centered at the origin of a coordinate plane. For any angle θ, the point where the terminal side of the angle intersects the unit circle has coordinates (cos(θ), sin(θ)). Therefore, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. When evaluating arccos(x), we are essentially looking for the angle on the unit circle within the range [0, π] whose x-coordinate is equal to x. This visual representation makes it easier to identify the angles corresponding to specific cosine values. For instance, consider the angle π/2. On the unit circle, this angle corresponds to the point (0, 1). Hence, cos(π/2) = 0, and arccos(0) = π/2. By understanding the relationship between angles and coordinates on the unit circle, you can efficiently navigate the inverse cosine function.

Now, let's tackle the specific problem of finding the value of arccos(-√3/2). The question we are addressing is: "What angle between 0 and π has a cosine of -√3/2?" To answer this, we can follow a systematic approach:

  1. Recall Special Angles: Begin by recalling the cosine values for common angles, particularly those in the first and second quadrants (0 to π). You should know that cos(π/6) = √3/2. However, we are looking for an angle with a negative cosine value. This tells us that the angle must lie in the second quadrant, where cosine is negative.
  2. Identify the Reference Angle: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is π/6, as cos(π/6) = √3/2. We are looking for an angle in the second quadrant with the same reference angle.
  3. Find the Angle in the Second Quadrant: To find the angle in the second quadrant with a reference angle of π/6, we subtract π/6 from π. This gives us π - π/6 = 5π/6. Therefore, the angle we are seeking is 5π/6.
  4. Verify the Solution: Double-check that cos(5π/6) = -√3/2 and that 5π/6 lies within the range of arccos, which is [0, π]. Since both conditions are met, we can confidently conclude that arccos(-√3/2) = 5π/6.

This step-by-step method allows you to systematically approach inverse trigonometric function problems. By combining your knowledge of special angles, reference angles, and the quadrants where trigonometric functions are positive or negative, you can accurately determine the values of inverse trigonometric expressions. Practice with various examples will further solidify your understanding and enhance your problem-solving skills.

Having determined the value of arccos(-√3/2) to be 5π/6, let's analyze the given options and understand why the others are incorrect:

  • A. π/6: This option represents an angle in the first quadrant where the cosine is positive (cos(π/6) = √3/2). Since we are looking for an angle with a negative cosine, this option is incorrect.
  • B. π/4: This option also represents an angle in the first quadrant with a positive cosine (cos(π/4) = √2/2). Therefore, it is not the correct answer.
  • C. 3π/4: While 3π/4 lies in the second quadrant where cosine is negative, cos(3π/4) = -√2/2, not -√3/2. This option is incorrect.
  • D. 5π/6: As we previously calculated, cos(5π/6) = -√3/2, and 5π/6 lies within the range of arccos [0, π]. Thus, this is the correct answer.

By carefully considering the cosine values of the given angles and their respective quadrants, we can confidently eliminate the incorrect options and pinpoint the correct solution. This process not only reinforces your understanding of inverse trigonometric functions but also enhances your test-taking strategies.

In conclusion, the value of arccos(-√3/2) is 5π/6. This determination involves understanding the definition of the inverse cosine function, its range and domain, the unit circle representation, and a systematic approach to finding angles with specific cosine values. By recalling special angles, utilizing reference angles, and considering the quadrants where cosine is positive or negative, you can efficiently evaluate arccos expressions.

The inverse cosine function, arccos(x), seeks the angle whose cosine is x, restricted to the range [0, π]. It's essential to remember that the output of arccos(x) will always be an angle between 0 and π radians. The unit circle provides a powerful visual tool for understanding the relationship between angles and their cosine values. When evaluating arccos(-√3/2), we look for the angle in the second quadrant (where cosine is negative) that has a reference angle of π/6. This leads us to the correct answer, 5π/6.

Further exploration of inverse trigonometric functions can involve solving more complex trigonometric equations, graphing inverse trigonometric functions, and applying these concepts in real-world scenarios. Consider exploring other inverse trigonometric functions such as arcsin(x) and arctan(x) and their respective properties. Understanding these functions is crucial for various applications in physics, engineering, and computer science. Practice with a variety of problems will solidify your understanding and enhance your ability to apply these concepts effectively.

By mastering the inverse cosine function and related concepts, you will not only strengthen your trigonometric foundation but also develop valuable problem-solving skills applicable across various disciplines. Embrace the challenge, explore the nuances, and unlock the power of inverse trigonometric functions.