Calculating Angle BAC Measure Using Inverse Sine Function

In this article, we will delve into the problem of calculating the measure of angle BAC using the given equation sin⁻¹(3.1/4.5) = x. This problem falls under the domain of trigonometry, a branch of mathematics that deals with the relationships between the sides and angles of triangles. Understanding trigonometric functions and their inverses is crucial for solving such problems. This article aims to provide a comprehensive explanation of the solution, ensuring clarity and a deep understanding of the underlying concepts.

Let's embark on a journey to unravel the mystery of angle BAC. We're given the equation sin⁻¹(3.1/4.5) = x, which looks like a complex trigonometric puzzle. But don't worry, we'll break it down step by step to reveal its secrets. The heart of this problem lies in understanding inverse trigonometric functions, particularly the arcsine (sin⁻¹) function. Think of it as the counterpart to the regular sine function. While the sine function tells us the ratio of the opposite side to the hypotenuse in a right triangle, the arcsine function does the reverse – it tells us the angle whose sine is a given ratio. In our case, we're looking for the angle x whose sine is 3.1/4.5. Imagine a right triangle where the side opposite angle BAC has a length of 3.1 units, and the hypotenuse (the side opposite the right angle) has a length of 4.5 units. The arcsine function helps us find the measure of angle BAC in this triangle. To solve this, we'll use a calculator to find the arcsine of 3.1/4.5. Make sure your calculator is set to degree mode, as we want the answer in degrees. The result will give us the measure of angle BAC. Once we have the value, we'll round it to the nearest whole degree as requested in the problem. This process transforms an abstract equation into a concrete angle measurement, bridging the gap between mathematical symbols and geometric reality. By understanding the relationship between the sides and angles of a triangle, we can confidently solve trigonometric problems and appreciate the power of these functions in describing the world around us.

Step-by-Step Solution

1. Understand the equation: The equation sin⁻¹(3.1/4.5) = x tells us that we need to find the angle x whose sine is equal to 3.1/4.5. In other words, if we take the sine of angle x, we should get 3.1/4.5.
2. Calculate the ratio: First, we need to calculate the value of the fraction 3.1/4.5. This gives us approximately 0.6889.
3. Use the inverse sine function: Now, we need to find the angle whose sine is 0.6889. This is where the inverse sine function (sin⁻¹, also known as arcsine) comes in. We use a calculator to find sin⁻¹(0.6889).
4. Calculator usage: Make sure your calculator is in degree mode. Enter sin⁻¹(0.6889) into your calculator. The result will be approximately 43.55 degrees.
5. Round to the nearest whole degree: The problem asks us to round the answer to the nearest whole degree. Since 43.55 is closer to 44 than 43, we round up to 44 degrees.

Answer

Therefore, the measure of angle BAC, rounded to the nearest whole degree, is 44°. So, the correct answer is C. 44°.

Distractor Analysis

It's important to understand why the other options are incorrect. This helps to solidify your understanding of the concept and avoid similar mistakes in the future.

• A. 0°: This is incorrect because the sine of 0° is 0, which is not equal to 3.1/4.5. An angle of 0 degrees would mean that the triangle is essentially a straight line, with no height for the opposite side.
• B. 1°: This is also incorrect. The sine of 1° is a very small value (approximately 0.017), which is far from 0.6889. This option might be a result of a misunderstanding of the scale of angles and their sine values.

Why 44° is the Correct Answer

The angle 44° is the correct answer because its sine is approximately 0.6947, which is very close to the calculated ratio of 3.1/4.5 (0.6889). The inverse sine function accurately gives us the angle that corresponds to this ratio. This demonstrates the practical application of trigonometric functions in determining angles within geometric figures.

To fully grasp the solution and tackle similar problems, understanding some core trigonometric concepts is essential. Let's explore these concepts in detail. Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It's particularly focused on right triangles, which are triangles containing one angle of 90 degrees. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate an angle in a right triangle to the ratios of its sides. 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 cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. And the tangent of an angle is the ratio of the opposite side to the adjacent side. Remember the acronym SOH CAH TOA to easily recall these relationships: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. These functions are fundamental for calculating unknown sides or angles in triangles. But what if we know the ratio and want to find the angle? That's where inverse trigonometric functions come into play. The inverse trigonometric functions, also known as arc functions, are the inverses of the sine, cosine, and tangent functions. They are denoted as sin⁻¹, cos⁻¹, and tan⁻¹, respectively. For example, if sin(x) = y, then sin⁻¹(y) = x. These functions allow us to find the angle when we know the ratio of the sides. The inverse sine function (sin⁻¹) gives us the angle whose sine is a given value. Similarly, the inverse cosine function (cos⁻¹) gives us the angle whose cosine is a given value, and the inverse tangent function (tan⁻¹) gives us the angle whose tangent is a given value. In our problem, we used the inverse sine function to find the angle BAC because we were given the ratio of the opposite side to the hypotenuse. Understanding these fundamental trigonometric concepts is crucial for solving a wide range of problems in geometry, physics, and engineering.

Inverse Trigonometric Functions: The Key to Unlocking Angles

Dive deeper into inverse trigonometric functions, the unsung heroes that allow us to find angles from ratios. Inverse trigonometric functions are essential tools in trigonometry, allowing us to