Finding Angle LKJ Solving Tan-1(8.9 7.7) Equation

Jul 12, 2025 by Admin 50 views

Finding the Measure of Angle LKJ Using Inverse Tangent

In trigonometry, the inverse tangent function, denoted as $\tan^{-1}(x)$ or arctan $(x)$ , plays a crucial role in determining angles when the ratio of the opposite side to the adjacent side in a right-angled triangle is known. This article delves into the application of the inverse tangent function to find the measure of an angle, specifically angle LKJ, given the equation $\tan^{-1}(\frac{8.9}{7.7}) = x$ . We will explore the steps involved in solving this equation, understand the underlying trigonometric principles, and arrive at the solution, rounding it to the nearest whole degree.

The tangent function, in basic trigonometry, is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. The inverse tangent function, conversely, takes this ratio as input and returns the angle whose tangent is that ratio. Mathematically, if $\tan(\theta) = y$ , then $\tan^{-1}(y) = \theta$ . The inverse tangent function is particularly useful when we know the ratio of sides in a right triangle but need to find the angle itself.

When using a calculator to find the inverse tangent, it's important to ensure that the calculator is set to the correct mode – degrees or radians – depending on the desired unit of measurement for the angle. In most practical applications, especially in geometry and navigation, angles are measured in degrees.

Given the equation $\tan^{-1}(\frac{8.9}{7.7}) = x$ , our goal is to find the value of $x$ , which represents the measure of angle LKJ. To do this, we need to evaluate the inverse tangent of the fraction $\frac{8.9}{7.7}$ .

First, we calculate the numerical value of the fraction:

$\frac{8.9}{7.7} \approx 1.1558$

Next, we use the inverse tangent function to find the angle whose tangent is approximately 1.1558. Using a calculator, we input $\tan^{-1}(1.1558)$ and obtain the result:

$x \approx 49.14^{\circ}$

Since the question requires us to round the answer to the nearest whole degree, we round 49.14° to 49°.

Therefore, the measure of angle LKJ is approximately 49 degrees. This demonstrates a practical application of the inverse tangent function in determining angles from given ratios.

To solve the equation $\tan^{-1}(\frac{8.9}{7.7}) = x$ and find the measure of angle LKJ, we follow these detailed steps:

Understand the Problem: The equation implies that the tangent of angle $x$ (angle LKJ) is equal to the ratio $\frac{8.9}{7.7}$ . The inverse tangent function will help us find the angle $x$ .
Calculate the Ratio: Divide 8.9 by 7.7 to find the numerical value of the ratio. This value represents the tangent of the angle we are trying to find. $\frac{8.9}{7.7} \approx 1.1558$
Apply the Inverse Tangent Function: Use a calculator to compute the inverse tangent (arctan) of the ratio obtained in the previous step. Ensure your calculator is set to degree mode, as we need the angle in degrees. $x = \tan^{-1}(1.1558)$ Using a calculator: $x \approx 49.14^{\circ}$
Round to the Nearest Whole Degree: Round the result to the nearest whole degree as required by the problem statement. $x \approx 49^{\circ}$

Thus, the measure of angle LKJ is approximately 49 degrees. The steps above provide a clear, methodical approach to solving such trigonometric problems.

Rounding is a crucial aspect of mathematical calculations, especially in trigonometry, where decimal values often arise. The degree of rounding depends on the context of the problem and the required level of precision. In this case, the problem explicitly asks for the answer to be rounded to the nearest whole degree.

Rounding ensures that the answer is presented in a simplified and easily understandable form. It also acknowledges the limitations of measurement and the potential for minor inaccuracies in real-world applications. For instance, when measuring angles in construction or navigation, rounding to the nearest degree might be sufficient for practical purposes.

However, it's important to note that excessive rounding can lead to significant errors in certain situations. Therefore, it's essential to follow the instructions provided in the problem statement and exercise caution when rounding intermediate values in multi-step calculations. In this problem, we rounded only the final answer to maintain accuracy throughout the calculation process. This demonstrates the importance of precision while adhering to rounding guidelines.

The inverse tangent function has numerous practical applications in various fields, including:

Navigation: Determining bearings and headings in navigation systems.
Engineering: Calculating angles in structural designs and mechanical systems.
Physics: Finding angles of projectile motion and vector components.
Computer Graphics: Rotating objects and calculating viewing angles in 3D graphics.
Surveying: Measuring angles and distances in land surveying.

In each of these applications, the inverse tangent function allows us to find angles when we know the ratio of two sides in a right-angled triangle. For example, in navigation, if a ship's east-west and north-south displacement is known, the bearing (angle relative to north) can be calculated using the inverse tangent function. The versatility of the inverse tangent function makes it an indispensable tool in many technical disciplines.

In summary, the equation $\tan^{-1}(\frac{8.9}{7.7}) = x$ is used to find the measure of angle LKJ. By calculating the inverse tangent of the given ratio and rounding to the nearest whole degree, we determined that angle LKJ is approximately 49 degrees. This exercise highlights the practical application of the inverse tangent function in solving real-world problems involving angles and trigonometric ratios. The detailed steps and explanations provided in this article offer a comprehensive understanding of the process, emphasizing the importance of accurate calculations and appropriate rounding techniques. Mastering the use of inverse trigonometric functions is essential for anyone working in fields that require precise angle measurements and calculations.

Therefore, the correct answer is C. 49°