Arccos(-√3/2): Find The Exact Value

Hey guys! Let's dive into a common trigonometry problem: figuring out the exact value of arccos(-√3/2). This might seem tricky at first, but with a good grasp of inverse trigonometric functions and the unit circle, it becomes super manageable. So, let's break it down step by step and make sure we understand not just the how, but also the why behind the answer.

Understanding Arccosine

Before we jump into solving arccos(-√3/2), it's essential to understand what arccosine actually means. Arccosine, written as arccos(x) or cos⁻¹(x), is the inverse function of cosine. Think of it this way: cosine takes an angle and gives you a ratio (the x-coordinate on the unit circle), while arccosine takes a ratio and gives you the angle. The key here is the range of arccosine. By definition, the arccosine function only outputs angles between 0 and π radians (0° and 180°). This restriction is super important because the cosine function is periodic, meaning it repeats its values. Without this restriction, arccosine wouldn't be a well-defined function. So, when we're looking for arccos(-√3/2), we're searching for an angle within the interval [0, π] whose cosine is -√3/2. We need to remember this as we proceed through solving the problem.

To really nail down what arccosine is about, let’s consider why it has this limited range. The cosine function, when you look at its full graph, goes up and down forever, repeating its values. This means that if you just asked, "Hey, what angle has a cosine of, say, 0.5?" there would be infinitely many answers! To make arccosine useful as a true function (where each input has only one output), we chop the cosine function down to a piece that passes the horizontal line test. This piece is the part between 0 and π radians. This way, for any value between -1 and 1 that you feed into arccosine, you get back exactly one angle in that range. This makes arccosine predictable and useful in all sorts of calculations and applications. So, keeping this range in mind, 0 to π radians, will help us find the one correct answer for arccos(-√3/2).

Using the Unit Circle

The unit circle is our best friend when it comes to trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane. The x-coordinate of any point on the unit circle represents the cosine of the angle formed between the positive x-axis and the line connecting the origin to that point. Similarly, the y-coordinate represents the sine of the angle. To find arccos(-√3/2), we need to find the angle on the unit circle between 0 and π where the x-coordinate (cosine) is -√3/2. Now, think about the special angles we know well: 30°, 45°, and 60° (or π/6, π/4, and π/3 radians). We know that cos(30°) = √3/2, cos(45°) = √2/2, and cos(60°) = 1/2. Since we're looking for -√3/2, we need to be in the second quadrant where cosine values are negative. In the second quadrant, the reference angle that corresponds to 30° (π/6 radians) is 150° (5π/6 radians). This is because 180° - 30° = 150°, or π - π/6 = 5π/6.

The unit circle isn't just a visual aid; it's a powerhouse of trigonometric information. It allows us to quickly recall the sine and cosine values of common angles, making problems like arccos(-√3/2) much easier to tackle. When you think about it, the unit circle neatly organizes all the possible angles and their corresponding ratios into one easily digestible diagram. The symmetry inherent in the circle also helps us find related angles. For instance, once we know the cosine and sine of an angle in the first quadrant, we can easily figure out the cosine and sine of its counterparts in the other quadrants simply by changing the signs appropriately. For arccos(-√3/2), recognizing that the cosine value is negative immediately tells us we're looking for an angle in either the second or third quadrant. However, since the range of arccosine is [0, π], we can narrow our search down to just the second quadrant. This kind of logical deduction, guided by the unit circle, is what makes these problems feel less like rote memorization and more like a puzzle to be solved. So, keeping the unit circle in mind is crucial for mastering trigonometry and inverse trigonometric functions.

The Solution

As we discussed, we're searching for an angle θ in the range [0, π] such that cos(θ) = -√3/2. Looking at the unit circle and considering the special angles, we know that the angle 5π/6 (or 150°) satisfies this condition. Cos(5π/6) = -√3/2, and 5π/6 falls within the acceptable range for arccosine (0 to π). Therefore, the value of arccos(-√3/2) is 5π/6.

To solidify this understanding, let’s walk through why 5π/6 is the definitive answer. First, we've already established that the arccosine function has a restricted range of 0 to π radians. This means we're only looking for solutions within the top half of the unit circle. When we identify that the cosine value is -√3/2, we know we're dealing with an angle that's related to the 30-60-90 triangle, since √3/2 is a common value associated with these triangles. The reference angle whose cosine is √3/2 is π/6 (30°). But because our cosine value is negative, we need an angle in the second quadrant that has the same reference angle. This is where we get 5π/6. It's π - π/6, which places it in the second quadrant, and its cosine is indeed -√3/2. No other angle within the range of 0 to π satisfies this condition. If we were to go further around the circle, past π, we'd find another angle with a cosine of -√3/2, but that angle would be outside the defined range of arccosine. Thus, 5π/6 is the one and only correct answer, perfectly fitting both the trigonometric requirements and the functional definition of arccosine. This careful consideration of range and reference angles is what allows us to confidently solve these kinds of problems.

Conclusion

So, to wrap things up, the value of arccos(-√3/2) is 5π/6 radians, or 150 degrees. Remember, understanding the concept of inverse trigonometric functions and using the unit circle are key to solving these problems. Keep practicing, and you'll nail these in no time! Understanding arccosine and the unit circle opens doors to more complex trig problems and applications, so it’s a great concept to really master. Keep exploring and have fun with math, guys! We've successfully navigated this trigonometric challenge by breaking it down into manageable steps: understanding the definition and range of arccosine, utilizing the unit circle to visualize angles and cosine values, and finally, pinpointing the exact angle that satisfies our conditions. This methodical approach is crucial for tackling similar problems and building a strong foundation in trigonometry. So next time you encounter an inverse trigonometric function, remember to think about the range restrictions and how the unit circle can guide you to the solution. With practice, these types of questions will become second nature!