Calculating Angle BAC Using Inverse Sine A Comprehensive Guide

Calculate the measure of angle BAC using the equation $\sin ^{-1}(\frac{3.1}{4.5})=x$. Round the answer to the nearest whole degree.

In trigonometry, finding the measure of angles is a common task, especially in scenarios involving triangles. The given problem presents a scenario where we need to calculate the measure of angle BAC using the equation $\sin^{-1}(\frac{3.1}{4.5}) = x$. This involves the use of the inverse sine function, also known as arcsin, which is a fundamental concept in trigonometry. In this comprehensive guide, we will delve into the step-by-step process of calculating the angle BAC, understand the underlying principles, and explore the significance of this calculation in various practical applications. We will break down the equation, explain the concept of inverse trigonometric functions, and provide a detailed solution to determine the measure of angle BAC, rounding it to the nearest whole degree as required. Understanding how to apply trigonometric functions and their inverses is crucial for anyone studying mathematics, physics, engineering, or related fields. This article aims to provide a clear and thorough explanation, making it easier for students and enthusiasts to grasp the concepts and confidently tackle similar problems.

Understanding the Problem

The problem states that the measure of angle BAC can be calculated using the equation $\sin^{-1}(\frac{3.1}{4.5}) = x$. This equation tells us that the angle x, which is the measure of angle BAC, is the angle whose sine is $\frac{3.1}{4.5}$. To find the value of x, we need to use the inverse sine function ($\sin^{-1}$), also known as arcsin. The inverse sine function is the inverse of the sine function, and it helps us find the angle when we know the sine value. In mathematical terms, if $\sin(x) = y$, then $\sin^{-1}(y) = x$. This relationship is fundamental to understanding how to solve this type of problem. The given equation can be visualized as a right-angled triangle where the ratio of the opposite side to the hypotenuse is $\frac{3.1}{4.5}$. The angle x is the angle opposite the side of length 3.1 in a right-angled triangle with a hypotenuse of length 4.5. To solve this, we will first calculate the value inside the inverse sine function, which is $\frac{3.1}{4.5}$, and then use a calculator to find the inverse sine of that value. This will give us the measure of angle BAC in degrees. Understanding the context and the mathematical principles involved is crucial before proceeding with the calculation. This foundational knowledge will ensure that the solution is not only correct but also understood conceptually.

Step-by-Step Solution

To find the measure of angle BAC, we will follow a step-by-step approach using the given equation $\sin^{-1}(\frac{3.1}{4.5}) = x$. This process involves evaluating the fraction inside the inverse sine function and then applying the inverse sine function to find the angle in degrees. First, we calculate the value of the fraction $\frac{3.1}{4.5}$. Performing this division, we get: $\frac{3.1}{4.5} ≈ 0.6889$. This value represents the sine of the angle we are trying to find. Next, we use the inverse sine function ($\sin^{-1}$) to find the angle whose sine is approximately 0.6889. This is where a calculator becomes essential. Most scientific calculators have an inverse sine function, often labeled as sin⁻¹ or arcsin. Using a calculator, we input the value 0.6889 and apply the inverse sine function: $\sin^{-1}(0.6889) ≈ 43.55$ degrees. Finally, we round the result to the nearest whole degree as required by the problem. Rounding 43.55 degrees to the nearest whole degree gives us 44 degrees. Therefore, the measure of angle BAC is approximately 44 degrees. This step-by-step solution breaks down the process into manageable parts, making it easier to follow and understand. Each step is crucial in arriving at the correct answer, from the initial calculation of the fraction to the final rounding of the result.

Detailed Calculation

The detailed calculation of the angle BAC involves a series of steps to ensure accuracy and a clear understanding of the process. Let's break down each step further. First, we need to evaluate the fraction inside the inverse sine function: $\frac{3.1}{4.5}$. To do this, we perform the division: 3.1 ÷ 4.5. The result is approximately 0.6888888... For practical purposes, we can round this value to four decimal places, which gives us 0.6889. This fraction represents the ratio of the side opposite angle BAC to the hypotenuse in a right-angled triangle. Next, we apply the inverse sine function ($\sin^{-1}$) to the value we just calculated. The inverse sine function gives us the angle whose sine is the given value. In this case, we want to find the angle x such that $\sin(x) ≈ 0.6889$. Using a scientific calculator, we input the value 0.6889 and press the inverse sine button (usually labeled as sin⁻¹ or arcsin). The calculator will return a value in degrees. The result we obtain is approximately 43.55 degrees. This value represents the measure of angle BAC in degrees. Finally, we need to round the result to the nearest whole degree. The problem statement specifically asks for the answer to be rounded to the nearest whole degree. Since the decimal part of 43.55 is 0.55, which is greater than or equal to 0.5, we round up to the next whole number. Therefore, 43.55 degrees rounded to the nearest whole degree is 44 degrees. This detailed calculation provides a thorough understanding of each step, ensuring that the final answer is accurate and well-understood.

Practical Applications

The calculation of angles using inverse trigonometric functions, such as the one we performed for angle BAC, has numerous practical applications across various fields. In engineering, these calculations are crucial for designing structures, machines, and electrical circuits. For example, engineers use trigonometric functions to determine the angles and forces involved in structural supports, ensuring the stability and safety of buildings and bridges. In physics, inverse trigonometric functions are essential for analyzing projectile motion, wave phenomena, and optics. Calculating angles of incidence and refraction, as well as determining the direction and magnitude of forces, often involves the use of arcsin, arccos, and arctan. In navigation, these functions are used to determine the direction and distance between two points. Pilots and sailors use trigonometry to calculate their course, taking into account factors such as wind and current. In computer graphics and game development, trigonometric functions are used extensively to create realistic movements and interactions. Angles and rotations are fundamental to creating 3D models, animations, and simulations. Even in everyday life, the principles of trigonometry are applied, often without conscious thought. For example, when parking a car, estimating the height of a tree, or angling a ladder against a wall, we are implicitly using trigonometric concepts. Furthermore, in surveying and mapping, accurate angle measurements are critical for creating detailed maps and property boundaries. Surveyors use instruments like theodolites and total stations to measure angles and distances, which are then used to create accurate representations of the terrain. In summary, the ability to calculate angles using inverse trigonometric functions is a fundamental skill with wide-ranging applications. From constructing buildings to navigating the seas, these calculations are essential for solving real-world problems and advancing our understanding of the world around us.

Conclusion

In conclusion, we have successfully calculated the measure of angle BAC using the equation $\sin^{-1}(\frac{3.1}{4.5}) = x$. By following a step-by-step approach, we first evaluated the fraction $\frac{3.1}{4.5}$, then applied the inverse sine function to find the angle, and finally rounded the result to the nearest whole degree. The result, as we found, is approximately 44 degrees. This process highlights the importance of understanding and applying inverse trigonometric functions in solving real-world problems. The ability to calculate angles using these functions is crucial in various fields, including engineering, physics, navigation, and computer graphics. Moreover, the detailed explanation provided in this article demonstrates the practical application of trigonometry in everyday scenarios. From constructing buildings to navigating ships, the principles of trigonometry are fundamental to many aspects of our lives. By mastering these concepts, students and professionals alike can enhance their problem-solving skills and gain a deeper appreciation for the mathematical principles that govern the world around us. Therefore, understanding and practicing these calculations is not only academically beneficial but also essential for practical applications in numerous professions and daily activities. The step-by-step approach outlined in this guide serves as a valuable resource for anyone looking to improve their understanding of trigonometry and its applications. This comprehensive exploration underscores the significance of mathematical literacy in navigating and understanding the complexities of our world.

Answer: C. 44°