Calculate Angle BAC Measure Step-by-Step Solution And Explanation

In mathematics, determining angles is a fundamental skill with applications across various fields, from geometry and trigonometry to physics and engineering. This article delves into the process of calculating the measure of angle BAC using the given equation and provides a comprehensive understanding of the underlying principles. We will explore the concept of inverse trigonometric functions, specifically the arcsine function, and demonstrate how to apply it to solve for angles in geometric figures. Furthermore, we will discuss the importance of accurate calculations and rounding techniques to ensure precise results. This guide aims to equip you with the knowledge and skills necessary to confidently tackle similar problems involving angle measurements.

Understanding the Problem

In this problem, we are tasked with finding the measure of angle BAC, which can be calculated using the equation sin⁻¹(3.1/4.5) = x. This equation involves the inverse sine function, also known as arcsine, which is a crucial concept in trigonometry. The notation sin⁻¹ represents the inverse of the sine function. In simpler terms, it answers the question, "What angle has a sine of this value?" The fraction 3.1/4.5 represents the ratio of the side opposite angle BAC to the hypotenuse in a right-angled triangle. This ratio is the sine of angle BAC. The phrase "not drawn to scale" is a crucial reminder that we cannot rely on visual estimations from a diagram; instead, we must use the given equation to find the accurate measure of the angle. This emphasis on calculation over visual approximation is a core principle in mathematical problem-solving. Understanding this setup is the first step towards correctly solving for the angle BAC. We must grasp the relationship between the sine function, its inverse, and the sides of a triangle to proceed effectively. By carefully analyzing the given information and the equation, we can pave the way for a precise and accurate solution.

Inverse Trigonometric Functions

Inverse trigonometric functions, often called arcus functions, are the inverses of the basic trigonometric functions: sine, cosine, and tangent. These functions are essential for finding angles when we know the ratio of sides in a right triangle. Specifically, the arcsine function (sin⁻¹ or arcsin) gives us the angle whose sine is a given value. Similarly, arccosine (cos⁻¹ or arccos) gives the angle whose cosine is a given value, and arctangent (tan⁻¹ or arctan) gives the angle whose tangent is a given value. Understanding these functions is crucial for solving a wide range of problems in trigonometry and geometry. The arcsine function, which is relevant to our problem, takes a value between -1 and 1 as input and returns an angle between -90° and 90° (or -π/2 and π/2 radians). This range is important to remember because it defines the principal values of the arcsine function. When using a calculator to find the arcsine, it will always provide an angle within this range. The concept of inverse functions is fundamental in mathematics, and trigonometric functions are no exception. Each trigonometric function has a corresponding inverse function that "undoes" the operation of the original function. This allows us to work backward from a ratio to find the angle. In the context of our problem, we are given the ratio 3.1/4.5, which represents the sine of angle BAC. To find the angle itself, we need to apply the arcsine function. This underscores the importance of understanding and utilizing inverse trigonometric functions in solving geometric and trigonometric problems.

Applying the Arcsine Function

To calculate the measure of angle BAC using the equation sin⁻¹(3.1/4.5) = x, we need to apply the arcsine function to the ratio 3.1/4.5. This can be done using a scientific calculator. First, ensure your calculator is in degree mode, as the question asks for the answer to be rounded to the nearest whole degree. Then, input the expression "arcsin(3.1/4.5)" or "sin⁻¹(3.1/4.5)" into your calculator. The calculator will return the angle whose sine is 3.1/4.5. The result will be a decimal number, representing the angle in degrees. It's crucial to understand the order of operations when using a calculator. First, divide 3.1 by 4.5 to get the ratio. Then, apply the arcsine function to this ratio. This ensures that you are finding the angle whose sine is the calculated ratio. Using a calculator efficiently is a key skill in mathematics. Familiarize yourself with the functions on your calculator, especially the trigonometric and inverse trigonometric functions. Practice using these functions with different inputs to build confidence and accuracy. When dealing with inverse trigonometric functions, it's essential to pay attention to the range of possible outputs. The arcsine function, as mentioned earlier, returns angles between -90° and 90°. If your problem involves angles outside this range, you may need to consider additional solutions based on the properties of trigonometric functions. However, in this specific problem, we are looking for an angle within the standard range, making the direct application of the arcsine function straightforward.

Calculation and Rounding

After inputting the expression sin⁻¹(3.1/4.5) into a calculator, we obtain a result. The calculator should display a value approximately equal to 43.446 degrees. However, the question asks us to round the answer to the nearest whole degree. Rounding is a fundamental mathematical skill that involves approximating a number to a specified degree of accuracy. In this case, we need to round 43.446 to the nearest whole number. To round to the nearest whole degree, we look at the digit in the tenths place, which is 4 in this case. If this digit is 5 or greater, we round up the whole number. If it is less than 5, we round down. Since 4 is less than 5, we round down to 43 degrees. Therefore, the measure of angle BAC, rounded to the nearest whole degree, is 43°. It's important to understand the rules of rounding and apply them consistently. Rounding errors can accumulate if not done correctly, leading to inaccurate final answers. When rounding, always consider the context of the problem and the level of precision required. In some cases, rounding to the nearest whole number is sufficient, while in others, more decimal places may be necessary. In this problem, rounding to the nearest whole degree is appropriate as per the instructions. The process of calculation and rounding is an integral part of mathematical problem-solving. It's not enough to simply obtain a numerical result; we must also ensure that the result is presented in the correct format and with the appropriate level of precision.

Solution and Conclusion

Based on our calculations, the measure of angle BAC, rounded to the nearest whole degree, is 43°. Therefore, the correct answer is C. This solution demonstrates the application of inverse trigonometric functions, specifically the arcsine function, in solving for angles in geometric problems. We have shown how to use a calculator to find the arcsine of a ratio and how to round the result to the desired level of precision. The process involved understanding the problem, identifying the relevant trigonometric concept, applying the arcsine function, and rounding the result. Each step is crucial for arriving at the correct answer. This problem highlights the importance of a strong foundation in trigonometry and the ability to apply these concepts in practical situations. Understanding inverse trigonometric functions is essential for solving a wide range of problems in mathematics, physics, and engineering. By mastering these concepts, you can confidently tackle more complex problems involving angles and geometric figures. In conclusion, the measure of angle BAC is 43 degrees, obtained by correctly applying the arcsine function and rounding to the nearest whole degree. This solution underscores the significance of accurate calculations and a thorough understanding of trigonometric principles. The ability to solve such problems is a testament to one's mathematical proficiency and problem-solving skills.

What is the measure of angle BAC, rounded to the nearest whole degree, given that $\sin^{-1}(\frac{3.1}{4.5}) = x$ and the figure is not drawn to scale?

Calculate Angle BAC Measure Step-by-Step Solution and Explanation