Solving Arcsin Problems: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem that involves inverse sine functions (arcsin). This kind of stuff might seem a bit tricky at first, but trust me, with a little practice and a clear understanding of the concepts, you'll be acing these questions in no time. Today, we're going to break down the expression $\arcsin\left(-\frac{\sqrt{3}}{2}\right) + \arcsin\frac{1}{2} + \pi$. Don't worry, we'll go through it step by step, so you won't get lost along the way. Ready to get started? Let's go!

Understanding the Basics of Arcsin

Alright, before we jump into the problem, let's make sure we're all on the same page when it comes to arcsin. Arcsin, or the inverse sine function, is basically the opposite of the sine function. When we take the arcsin of a value, we're asking, "What angle has this sine value?" The arcsin function gives us an angle, and this angle is always within a specific range, usually from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ (or -90 degrees to 90 degrees). It's super important to remember this range because it helps us find the correct answer and avoid any confusion. Think of it like this: the sine function takes an angle and gives you a ratio (a number between -1 and 1), and the arcsin function takes that ratio and gives you the angle back. It is worth mentioning, that the arcsin of positive values gives us an angle in the first quadrant (0 to $\frac{\pi}{2}$), while the arcsin of negative values gives us an angle in the fourth quadrant ($-\frac{\pi}{2}$ to 0). This understanding is crucial for solving this type of problem. So, let’s go ahead and break down the given expression piece by piece. First up, we need to find the value of $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. What angle has a sine value of $-\frac{\sqrt{3}}{2}$? Because the value is negative, we know that the angle will be in the fourth quadrant. Also, we can recall the common angles from the unit circle, or our knowledge of special triangles such as the 30-60-90 triangle. The angle that has a sine of $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ or 60 degrees. Since the sine is negative, we know we need the angle in the fourth quadrant, so the answer is $-\frac{\pi}{3}$.

We know that the arcsin function is defined as the inverse of the sine function. The range of arcsin is limited to $[-\frac{\pi}{2}, \frac{\pi}{2}]$. When we calculate the arcsin function, we are essentially finding the angle (in radians) whose sine is equal to the input value. When the input value is positive, the arcsin function returns an angle in the first quadrant, while when the input value is negative, the arcsin function returns an angle in the fourth quadrant. This is something important that we will use to find the solution. Let's start with the first part of our problem: $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. We're looking for an angle whose sine is $-\frac{\sqrt{3}}{2}$. Remember that the sine function is negative in the fourth quadrant, so our angle will be negative. The reference angle for which the sine is $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$. Therefore, $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$.

So, arcsin is a fundamental concept in trigonometry, representing the inverse of the sine function. It's essentially about finding the angle corresponding to a given sine value. Remember, the output of arcsin (the angle) is always within a specific range, typically from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ radians, or -90 to 90 degrees. This range is super important because it helps us identify the correct solution. Now, let’s break down the given expression step-by-step. First, consider $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. This is asking: "What angle has a sine of $-\frac{\sqrt{3}}{2}$?" The negative sign indicates the angle is in the fourth quadrant (where sine is negative). From our knowledge of the unit circle or special triangles (like the 30-60-90 triangle), we know that the sine of 60 degrees (or $\frac{\pi}{3}$ radians) is $\frac{\sqrt{3}}{2}$. Since we need a negative sine value, we get $-\frac{\pi}{3}$ radians. We have successfully completed the first part of our expression.

Solving Each Part of the Expression

Now that we've refreshed our memories on arcsin, let's break down the given expression $\arcsin\left(-\frac{\sqrt{3}}{2}\right) + \arcsin\frac{1}{2} + \pi$ step by step. We'll find the value of each part and then add them up. This method allows us to simplify a complex expression step by step. This approach is helpful when dealing with more complex trigonometric expressions. Let's go through this process together, and it will become second nature in no time! First, we need to find $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. As we discussed earlier, this is the angle whose sine is $-\frac{\sqrt{3}}{2}$. Since the sine value is negative, we know the angle will be in the fourth quadrant. The angle whose sine is $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ (or 60 degrees). Therefore, $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$. Next, let's calculate $\arcsin\frac{1}{2}$. We are looking for an angle whose sine is $\frac{1}{2}$. The sine is positive, so the angle will be in the first quadrant. From our knowledge of special triangles or the unit circle, we know that $\sin\frac{\pi}{6} = \frac{1}{2}$ (or 30 degrees). So, $\arcsin\frac{1}{2} = \frac{\pi}{6}$. Finally, we just need to add $\pi$. Putting it all together, we have $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. Let's simplify this expression to find our final answer. Combining our results will allow us to arrive at our desired solution. The second part of our expression is $\arcsin\frac{1}{2}$. This is asking: "What angle has a sine of $\frac{1}{2}$?" The angle must be in the first quadrant (where the sine is positive). We know that $\sin\frac{\pi}{6} = \frac{1}{2}$ (30 degrees). So, $\arcsin\frac{1}{2} = \frac{\pi}{6}$. Now that we've evaluated each part, we need to add $\pi$. This is a straightforward addition, and we now know all the required components to solve this problem completely. The expression becomes $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. Let’s proceed to the next step, where we can add up our results to find the final answer. The third component of our expression is the constant $\pi$. Now we have all the components, so let’s substitute these values back into the original expression. Substituting the values we found, the expression becomes $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. This is a simple arithmetic problem, where we need to combine fractions with different denominators and add a constant. The key is to find a common denominator for the fractions. First, let's find the value of $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. This asks for the angle whose sine is $-\frac{\sqrt{3}}{2}$. The negative sign tells us the angle is in the fourth quadrant. The angle with a sine of $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$. So, $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$. Next, let's tackle $\arcsin\frac{1}{2}$. This is asking which angle has a sine of $\frac{1}{2}$. We know this is $\frac{\pi}{6}$. Finally, we have to add $\pi$. So, we have $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. Let's start with $-\frac{\pi}{3} + \frac{\pi}{6}$. To add these fractions, we need a common denominator, which is 6. So, $-\frac{\pi}{3}$ becomes $-\frac{2\pi}{6}$. Adding this to $\frac{\pi}{6}$, we get $-\frac{2\pi}{6} + \frac{\pi}{6} = -\frac{\pi}{6}$. Finally, we add $\pi$: $-\frac{\pi}{6} + \pi = -\frac{\pi}{6} + \frac{6\pi}{6} = \frac{5\pi}{6}$.

So, let's break down the expression step by step. First, we have $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. Remember that arcsin gives us an angle. We're looking for an angle whose sine is $-\frac{\sqrt{3}}{2}$. Since the sine is negative, we know the angle lies in the fourth quadrant. The reference angle whose sine is $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$. Therefore, $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$. Next, we have $\arcsin\frac{1}{2}$. This means we're looking for the angle whose sine is $\frac{1}{2}$. This angle is $\frac{\pi}{6}$. Finally, we add $\pi$. Putting it all together, we have $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. To solve this, we need to combine the fractions. The common denominator for $\frac{\pi}{3}$ and $\frac{\pi}{6}$ is 6. So, we rewrite $-\frac{\pi}{3}$ as $-\frac{2\pi}{6}$. Now we have $-\frac{2\pi}{6} + \frac{\pi}{6} = -\frac{\pi}{6}$. Adding $\pi$, we get $-\frac{\pi}{6} + \pi = -\frac{\pi}{6} + \frac{6\pi}{6} = \frac{5\pi}{6}$. Therefore, the final answer is $\frac{5\pi}{6}$. We need to calculate each part of the expression separately and then combine them. So, the original expression is $\arcsin\left(-\frac{\sqrt{3}}{2}\right) + \arcsin\frac{1}{2} + \pi$. First, we evaluate $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$. We know that the arcsin function returns the angle whose sine is equal to the given value. Here, the sine is negative, so the angle must be in the fourth quadrant. The reference angle for which $\sin$ is $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$. Therefore, $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$. Next, we calculate $\arcsin\frac{1}{2}$. The sine value is positive, which means the angle lies in the first quadrant. We know that $\sin\frac{\pi}{6} = \frac{1}{2}$, so $\arcsin\frac{1}{2} = \frac{\pi}{6}$. Finally, we add $\pi$. So, our expression is now $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. Let's simplify this. To add or subtract fractions, they must have a common denominator. The least common denominator for 3 and 6 is 6. Convert $-\frac{\pi}{3}$ to $-\frac{2\pi}{6}$. Thus, our expression becomes $-\frac{2\pi}{6} + \frac{\pi}{6} + \pi = -\frac{\pi}{6} + \pi$. To add $\pi$, we rewrite it with the same denominator: $\pi = \frac{6\pi}{6}$. Therefore, $-\frac{\pi}{6} + \frac{6\pi}{6} = \frac{5\pi}{6}$. The result of the whole expression will be equal to $\frac{5\pi}{6}$.

Putting it All Together and Finding the Answer

Now, let's put all the pieces together and find our answer. We've already calculated: $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$, $\arcsin\frac{1}{2} = \frac{\pi}{6}$, and we have to add $\pi$. Now, let's solve the whole thing. We have $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. First, we have to find a common denominator for the fractions. The least common denominator for 3 and 6 is 6. So, we convert $-\frac{\pi}{3}$ to $-\frac{2\pi}{6}$. Our equation becomes $-\frac{2\pi}{6} + \frac{\pi}{6} + \pi$. Combining the fractions, we get $-\frac{2\pi}{6} + \frac{\pi}{6} = -\frac{\pi}{6}$. Now we have $-\frac{\pi}{6} + \pi$. We rewrite $\pi$ with a denominator of 6, which is $\frac{6\pi}{6}$. So we have $-\frac{\pi}{6} + \frac{6\pi}{6} = \frac{5\pi}{6}$. And there you have it, guys! The final answer is $\frac{5\pi}{6}$. Let’s recap the steps we took and ensure we’ve found the correct result. First, we found the value of $\arcsin\left(-\frac{\sqrt{3}}{2}\right)$, which is $-\frac{\pi}{3}$. Then, we calculated $\arcsin\frac{1}{2}$, which is $\frac{\pi}{6}$. After that, we combined the fractions and added \pi. After these calculations, we've arrived at the final answer, which is $\frac{5\pi}{6}$.

To find the final answer, we first found that $\arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$ and $\arcsin\frac{1}{2} = \frac{\pi}{6}$. Then we add $\pi$ to the sum of the two previous results. So, we need to calculate $-\frac{\pi}{3} + \frac{\pi}{6} + \pi$. The least common multiple of 3 and 6 is 6. Therefore, let’s rewrite the expression with the common denominator of 6. We have $-\frac{2\pi}{6} + \frac{\pi}{6} + \frac{6\pi}{6} = \frac{-2\pi + \pi + 6\pi}{6} = \frac{5\pi}{6}$. Therefore, the final answer is $\frac{5\pi}{6}$. This process involves converting the fractions to a common denominator to simplify the addition and subtraction. Once we have combined all the terms, we will arrive at our final answer. Thus, our correct answer is $\frac{5\pi}{6}$. We can then look at the answer choices provided. Option 1 is $-\frac{5\pi}{6}$. Option 2 is $-\frac{2\pi}{3}$. Option 3 is $\frac{\pi}{2}$. Option 4 is $\frac{5\pi}{6}$. Option 5 is $\frac{2\pi}{3}$. The correct answer is Option 4: $\frac{5\pi}{6}$.

So, we've broken down this arcsin problem and found the solution. Always remember to understand the properties of the arcsin function, especially its range, and break the problem down into smaller, manageable steps. You'll be acing these questions in no time! Keep practicing, and you'll get better and better. Good job, guys! Keep up the great work! Remember, the key to solving these types of problems is to understand the properties of the arcsin function, including its range, and to break down the problem into smaller, manageable steps. By doing so, you can approach the problem logically and efficiently. The goal is to apply our knowledge of the unit circle, special triangles, and the properties of arcsin to correctly evaluate the expression. Through practice and a good understanding of these concepts, you'll be well-equipped to tackle any arcsin problem that comes your way. So, keep practicing, keep learning, and keep asking questions. You've got this!