Calculating Angle BAC Measure Using Inverse Sine Function

Calculating angles is a fundamental aspect of trigonometry, and understanding how to use inverse trigonometric functions is crucial for solving various problems. In this article, we will delve into how to find the measure of angle BAC using the given equation ${\sin^{-1}(\frac{3.1}{4.5}) = x}$. This problem involves the application of the inverse sine function, often denoted as arcsin or ${\sin^{-1}}$, to determine the angle whose sine is a specific value. Let’s break down the problem, understand the concepts involved, and arrive at the solution, rounding to the nearest whole degree.

Understanding the Basics of Inverse Sine

To truly grasp the concept of finding the measure of angle BAC, it's essential to begin with understanding the inverse sine function. The inverse sine, or arcsin, is the inverse operation of the sine function. In simpler terms, if ${\sin(x) = y}$, then ${\sin^{-1}(y) = x}$. The sine function relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse. The inverse sine function, on the other hand, takes a ratio (between -1 and 1) as input and returns the angle whose sine corresponds to that ratio. When we encounter an equation like ${\sin^{-1}(\frac{3.1}{4.5}) = x}$, it means we are looking for the angle x whose sine is ${\frac{3.1}{4.5}}$. This is a common scenario in various fields, including navigation, engineering, and physics, where angles need to be determined based on known ratios of sides in triangles. Therefore, mastering the use of the inverse sine is not just an academic exercise but a practical skill that can be applied in numerous real-world situations. Understanding the domain and range of the arcsin function is also crucial. The domain of ${\sin^{-1}(y)}$ is ${[-1, 1]}$, because the sine function's output always falls within this range. The principal range of the ${\sin^{-1}}$ function is ${[-\frac{\pi}{2}, \frac{\pi}{2}]}$, or -90° to 90°. This means that the calculator will give you an angle within this range. If you need an angle outside this range, you might need to consider the properties of the sine function in different quadrants. The inverse sine function is a fundamental tool in trigonometry that allows us to find angles when we know the ratio of the opposite side to the hypotenuse in a right triangle. Its applications extend beyond the classroom, making it a vital concept to understand for various practical and theoretical applications.

Setting Up the Equation for Angle BAC

The heart of the problem lies in the equation ${\sin^{-1}(\frac{3.1}{4.5}) = x}$, which directly relates to finding the measure of angle BAC. This equation is a concise way of expressing that the angle x, which corresponds to angle BAC in the context of the problem, is the angle whose sine is ${\frac{3.1}{4.5}}$. The fraction ${\frac{3.1}{4.5}}$ represents the ratio of the side opposite angle BAC to the hypotenuse in a right-angled triangle, a crucial piece of information when applying trigonometric principles. Understanding how this equation is set up requires recognizing the connection between trigonometric ratios and angles in triangles. In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The inverse sine function allows us to reverse this process, finding the angle when we know the ratio. In this specific scenario, 3.1 might represent the length of the side opposite angle BAC, and 4.5 might represent the length of the hypotenuse. By setting up the equation in this way, we are essentially translating a geometric problem into a mathematical one that can be solved using trigonometric functions. This setup is a critical step in problem-solving because it bridges the gap between the visual representation of a triangle and the numerical calculations needed to find the angle. Therefore, a clear understanding of this setup is paramount for accurately determining the measure of angle BAC and for solving similar trigonometric problems in the future. When setting up the equation, it's also important to consider the units and context of the problem. Angles are typically measured in degrees or radians, and the desired unit should be specified. In this case, we are asked to round to the nearest whole degree, indicating that we should express our answer in degrees. Furthermore, the context of the problem, such as the specific triangle or geometric figure, may provide additional information or constraints that need to be taken into account when setting up and solving the equation. Understanding these nuances can help prevent errors and ensure that the solution is both mathematically correct and meaningful in the given context.

Solving for x Using the Inverse Sine Function

Now, to solve for x in the equation ${\sin^{-1}(\frac{3.1}{4.5}) = x}$, we utilize the inverse sine function. This is typically done using a calculator that has the ${\sin^{-1}}$ or arcsin function. The process involves inputting the ratio ${\frac{3.1}{4.5}}$ into the inverse sine function and obtaining the corresponding angle in degrees or radians, depending on the calculator's setting. The steps to solve for x are straightforward yet require attention to detail to ensure accuracy. First, calculate the ratio ${\frac{3.1}{4.5}}$, which gives us approximately 0.6889. Next, input this value into the calculator's inverse sine function. Make sure your calculator is set to degree mode if you want the answer in degrees, which is the case in this problem since we are asked to round to the nearest whole degree. Once you compute ${\sin^{-1}(0.6889)}$, you should get an angle value. This value represents the measure of angle BAC in degrees. The use of a calculator is almost essential for this step, as manual computation of the inverse sine of a decimal value is complex and impractical. Different calculators may have slightly different interfaces for accessing the inverse sine function, but the underlying principle remains the same. It is also worth noting that the inverse sine function has a specific range of output values, typically between -90° and 90° (or -${\frac{\pi}{2}}$ and ${\frac{\pi}{2}}$ radians). This is because the sine function is periodic, and without restricting the range, there would be infinitely many angles with the same sine value. Therefore, the calculator will provide the angle within this principal range that satisfies the equation. Understanding the calculator's output and its relation to the range of the inverse sine function is important for interpreting the result correctly. In some cases, you may need to consider additional solutions or adjustments based on the context of the problem. However, in this particular scenario, the direct output from the calculator should provide the correct measure of angle BAC.

Rounding the Result to the Nearest Whole Degree

After obtaining the value of x using the inverse sine function, the final step is to round the result to the nearest whole degree, as specified in the problem statement. This rounding ensures that the answer is presented in the desired format and level of precision. Rounding is a fundamental mathematical skill that involves adjusting a number to a nearby value based on a set of rules. In this case, we are rounding to the nearest whole degree, which means we are looking for the integer value that is closest to the computed angle. The standard rounding rule states that if the decimal part of the number is 0.5 or greater, we round up to the next whole number. If the decimal part is less than 0.5, we round down to the current whole number. For example, if the calculator gives us an angle of 43.6 degrees, we would round up to 44 degrees. Conversely, if the angle is 43.2 degrees, we would round down to 43 degrees. The process of rounding to the nearest whole degree provides a level of simplification and approximation that is often suitable for practical applications. In many real-world scenarios, precise angle measurements are not necessary, and a rounded value is sufficient. Additionally, rounding can make the answer easier to understand and communicate. However, it is important to be mindful of the potential for rounding errors, especially in multi-step calculations. Rounding at intermediate steps can sometimes lead to a significant deviation in the final result. Therefore, it is generally recommended to perform calculations with as much precision as possible and only round the final answer. In the context of this problem, rounding to the nearest whole degree is the last step, ensuring that the final answer is both accurate and presented in the requested format. By following this rounding procedure, we can confidently provide the measure of angle BAC to the nearest whole degree.

Final Answer for the Measure of Angle BAC

After performing the calculations and rounding, we arrive at the final answer for the measure of angle BAC. Using a calculator, we find that ${\sin^{-1}(\frac{3.1}{4.5}) \approx 43.4^ ext{o}}$. Rounding this to the nearest whole degree, we get 43 degrees. Therefore, the measure of angle BAC is approximately 43 degrees. This final answer represents the solution to the problem and provides a numerical value for the angle in question. It is the culmination of the steps we have taken, from understanding the inverse sine function to setting up the equation, solving for x, and rounding the result. The process of arriving at this answer highlights the importance of each step and the interconnectedness of mathematical concepts. The accurate calculation and rounding are crucial for providing a correct and meaningful solution. This answer can then be used in further calculations or applications, depending on the context of the problem. For instance, if we were working on a larger geometric problem involving multiple angles and sides, the measure of angle BAC could be a critical piece of information for finding other unknown quantities. Moreover, the process of solving this problem reinforces the understanding of trigonometric principles and their application in real-world scenarios. The ability to find angles using inverse trigonometric functions is a valuable skill in various fields, including engineering, physics, and navigation. Therefore, the final answer not only solves the immediate problem but also contributes to a broader understanding of mathematical problem-solving. The answer of 43 degrees for the measure of angle BAC is a precise and practical solution, demonstrating the power of trigonometry in quantifying geometric relationships.

In conclusion, determining the measure of angle BAC using the equation ${\sin^{-1}(\frac{3.1}{4.5}) = x}$ involves a clear understanding of the inverse sine function, careful calculation, and appropriate rounding. The final answer, 43 degrees, is a testament to the application of these principles and provides a concrete solution to the problem.