Find Angle LKJ: Solving Tan-1(8.9/7.7)

Let's dive into solving a trigonometric problem to find the measure of a specific angle. We are given the equation $\tan^{-1}(\frac{8.9}{7.7}) = x$, and our mission is to determine the value of $x$, which represents the measure of angle LKJ. We need to round our final answer to the nearest whole degree. Let's break down the process step by step.

Understanding the Inverse Tangent Function

The inverse tangent function, denoted as $\tan^{-1}(y)$ or arctan$(y)$, answers the question: "What angle has a tangent of $y$?" In simpler terms, if $\tan(x) = y$, then $\tan^{-1}(y) = x$. This function is crucial in various fields like physics, engineering, and, of course, mathematics. It allows us to find angles when we know the ratio of the opposite side to the adjacent side in a right triangle.

When we have $\tan^{-1}(\frac{8.9}{7.7}) = x$, it means we are looking for the angle $x$ whose tangent is $\frac{8.9}{7.7}$. To find this angle, we'll use a calculator that has the inverse tangent function.

Step-by-Step Solution

1. Calculate the ratio: First, we need to compute the value of the fraction inside the inverse tangent function. So, we divide 8.9 by 7.7: $\frac{8.9}{7.7} \approx 1.1558$

2. Apply the inverse tangent: Now that we have the ratio, we apply the inverse tangent function to this value. This will give us the angle $x$ in radians or degrees, depending on the calculator's mode. Make sure your calculator is set to degree mode for this problem. $x = \tan^{-1}(1.1558)$

3. Use a calculator: Using a calculator, input the inverse tangent of 1.1558. The result should be approximately: $x \approx 49.14^{\circ}$

4. Round to the nearest whole degree: Finally, we round this value to the nearest whole degree. Since 49.14 is closer to 49 than to 50, we round down to 49. $x \approx 49^{\circ}$

Therefore, the measure of angle LKJ is approximately $49^{\circ}$.

Why This Matters

Understanding trigonometric functions and their inverses is vital for solving real-world problems. Imagine you are designing a ramp for a building. You know the height the ramp needs to reach and the horizontal distance it can cover. Using the inverse tangent function, you can calculate the angle of the ramp to ensure it meets safety standards.

Similarly, in navigation, knowing the angles between different points can help determine the direction and distance to travel. Engineers use these principles to design bridges, buildings, and other structures, ensuring they are stable and safe.

Common Mistakes to Avoid

• Incorrect Calculator Mode: One of the most common mistakes is having the calculator in the wrong mode (radians instead of degrees, or vice versa). Always double-check the mode before performing calculations.
• Rounding Errors: Rounding too early in the calculation can lead to inaccurate results. It's best to keep as many decimal places as possible until the final step.
• Forgetting the Inverse Function: Confusing the tangent function with its inverse can lead to incorrect setups. Remember, the inverse tangent gives you the angle when you know the ratio of the sides.
• Misunderstanding the Question: Always read the question carefully. In this case, we needed to find the angle and round it to the nearest whole degree. Misreading the instructions can lead to providing the wrong type of answer.

Conclusion

By following the correct steps and understanding the principles of inverse trigonometric functions, we found that the measure of angle LKJ is approximately $49^{\circ}$. This involves calculating the ratio, applying the inverse tangent function, and rounding to the nearest whole degree. Remember to avoid common mistakes such as using the wrong calculator mode or rounding too early.

The correct answer is C. $49^{\circ}$.

For further reading and a deeper understanding of trigonometric functions, you can visit Khan Academy's Trigonometry Section.