Finding Side C With Law Of Cosines Given A, B And Angle C

Jul 10, 2025 by ADMIN 58 views

In trigonometry, the Law of Cosines is a powerful tool for solving triangles when you know two sides and the included angle (the angle between them), or when you know all three sides. This article will guide you through the process of using the Law of Cosines to find the length of a side of a triangle when given the lengths of the other two sides and the measure of the included angle. Specifically, we'll focus on finding side c, given sides a and b and the angle ${ \angle C }$ . This is a common problem in various fields, including surveying, navigation, and engineering, where determining distances and angles is crucial. Understanding the Law of Cosines and its applications can greatly enhance your problem-solving skills in these areas.

The Law of Cosines is a generalization of the Pythagorean theorem, relating the lengths of the sides of a triangle to the cosine of one of its angles. It's particularly useful when dealing with non-right triangles, where the basic trigonometric ratios (SOH CAH TOA) don't directly apply. The Law of Cosines provides a way to calculate the length of a side if you know the lengths of the other two sides and the angle opposite the unknown side. There are three forms of the Law of Cosines, each solving for a different side, but they are essentially the same formula rearranged. The formula we will be using to find side c is:

c^2 = a^2 + b^2 - 2ab * cos(C)

Where:

c is the length of the side opposite angle ${ \angle C }$ .
a and b are the lengths of the other two sides.
${ \angle C }$ is the angle opposite side c.

This formula is a cornerstone of trigonometry and is essential for solving various triangle-related problems. It allows us to bridge the gap between side lengths and angles, providing a versatile tool for calculations in different scenarios. The Law of Cosines is derived using basic trigonometric principles and the distance formula, and it’s a fundamental concept for anyone working with triangles and their properties. Understanding the origins and applications of the Law of Cosines can give you a deeper appreciation for its usefulness in various mathematical and real-world contexts. Whether you're calculating distances in surveying, determining angles in navigation, or solving geometrical problems, the Law of Cosines is an indispensable tool in your mathematical arsenal.

In this specific problem, we are given the following information:

Side a = 2.98 miles
Side b = 3.43 miles
Angle ${ \angle C }$ = 40.8 degrees

Our goal is to find the length of side c, which is opposite angle ${ \angle C }$ . This scenario perfectly fits the application of the Law of Cosines, as we have two sides and the included angle, allowing us to directly calculate the third side. The problem is a classic example of how the Law of Cosines can be used in real-world scenarios, such as determining distances in land surveying or navigation. By applying the formula, we can find the missing side length with precision. The given values represent a practical situation where direct measurement of side c might be difficult or impossible, highlighting the utility of trigonometric methods in solving geometrical problems. This problem serves as an excellent illustration of how mathematical tools can provide solutions to real-world challenges. Understanding the problem setup and the information provided is the first crucial step in successfully applying the Law of Cosines and finding the desired side length.

To find the length of side c, we will use the Law of Cosines formula: c^2 = a^2 + b^2 - 2ab * cos(C). Let's break down the solution step by step:

Step 1: Substitute the given values into the formula.

We are given a = 2.98 miles, b = 3.43 miles, and ${ \angle C }$ = 40.8 degrees. Substituting these values into the Law of Cosines formula, we get:

c^2 = (2.98)^2 + (3.43)^2 - 2(2.98)(3.43) * cos(40.8°)

This substitution is a crucial step as it sets up the equation that we will solve for c. Ensuring that the values are correctly placed in the formula is essential for an accurate solution. The equation now represents the specific problem we are addressing, with all the known quantities in their respective places. This step transforms the general formula into a concrete equation tailored to our particular triangle, paving the way for the subsequent calculations that will lead us to the value of c. Each term in the equation corresponds to a specific part of the triangle, and their correct placement is fundamental to the integrity of the solution.

Step 2: Calculate the squares of sides a and b.

Next, we need to calculate the squares of sides a and b:

a^2 = (2.98)^2 = 8.8804
b^2 = (3.43)^2 = 11.7649

These calculations simplify the equation and make it easier to work with. Squaring the side lengths is a direct application of the Law of Cosines formula and is necessary to proceed with the calculation. The resulting values, a^2 and b^2, represent the squares of the respective side lengths and will be used in the next steps to find c^2. Accuracy in these calculations is paramount, as any errors here will propagate through the rest of the solution. Double-checking these values can help ensure the final result is correct. This step is a straightforward arithmetic operation, but its importance cannot be overstated in the overall solution process.

Step 3: Calculate the cosine of angle C.

Now, we need to find the cosine of angle ${ \angle C }$ , which is 40.8 degrees:

cos(40.8°) ≈ 0.7566 (using a calculator)

The cosine of an angle is a trigonometric function that relates the angle to the ratio of the adjacent side to the hypotenuse in a right triangle. However, in the Law of Cosines, we use the cosine function to relate angles and sides in any triangle, not just right triangles. Using a calculator to find the cosine of 40.8 degrees gives us an approximate value of 0.7566. This value is crucial for the next step, where we will multiply it by the other terms in the equation. The accuracy of this value depends on the precision of the calculator used, and it's generally a good practice to use a calculator that provides several decimal places to minimize rounding errors. This step bridges the gap between the angular measurement and its numerical representation, allowing us to incorporate it into the side-length calculations.

Step 4: Calculate 2ab * cos(C).

Next, we multiply 2, a, b, and the cosine of ${ \angle C }$ :

2ab * cos(40.8°) = 2 * 2.98 * 3.43 * 0.7566 ≈ 15.484

This calculation combines the side lengths a and b with the cosine of the included angle ${ \angle C }$ . The result, approximately 15.484, is a key component of the Law of Cosines formula. This step brings together all the individual pieces we've calculated so far, preparing us to find c^2. The multiplication ensures that the relationship between the sides and the angle is properly accounted for in the equation. Accuracy in this step is vital, as it directly impacts the final result. The value obtained here represents a significant portion of the calculation and is essential for determining the length of side c.

Step 5: Substitute the calculated values back into the formula and solve for c^2.

Now, we substitute the calculated values back into the Law of Cosines formula:

c^2 = 8.8804 + 11.7649 - 15.484
c^2 = 5.1613

This step combines the previously calculated values to find c^2. By adding a^2 and b^2 and then subtracting the term 2ab * cos(C), we isolate c^2 on one side of the equation. The result, 5.1613, represents the square of the length of side c. This value is the culmination of all the previous calculations and brings us closer to finding the actual length of side c. Accuracy in this step is crucial, as it directly determines the final value of c. The arithmetic operation here is straightforward, but its significance in the overall solution cannot be overstated.

Step 6: Take the square root to find c.

Finally, we take the square root of c^2 to find the length of side c:

c = √5.1613 ≈ 2.272 miles

Taking the square root of c^2 gives us the length of side c. The square root operation is the final step in solving for c using the Law of Cosines. The result, approximately 2.272 miles, is the answer to the problem. This value represents the length of the side opposite angle ${ \angle C }$ in the triangle. Accuracy in this step is paramount, as it directly provides the final solution. The square root operation undoes the squaring we performed earlier, allowing us to find the actual side length. This final calculation completes the problem, providing a concrete answer to the original question. The value of c is now known, and we have successfully used the Law of Cosines to solve the problem.

Therefore, the length of side c, rounded to three decimal places, is approximately 2.272 miles. This result is the solution to the problem, providing the length of the side opposite the given angle ${ \angle C }$ . The process of finding c involved several steps, each crucial to the accuracy of the final answer. We began by substituting the given values into the Law of Cosines formula, then calculated the squares of the sides and the cosine of the angle. Next, we combined these values to find c^2, and finally, we took the square root to find c. The final answer, 2.272 miles, is a precise measurement of the side length and can be used in further calculations or applications involving this triangle. This solution demonstrates the power and utility of the Law of Cosines in solving triangle-related problems. The result is a concrete value that answers the original question and provides a complete solution to the problem.

In conclusion, we successfully found the length of side c using the Law of Cosines. By following a step-by-step approach, we substituted the given values into the formula, performed the necessary calculations, and arrived at the final answer of approximately 2.272 miles. This process demonstrates the effectiveness of the Law of Cosines in solving for unknown sides in triangles when given two sides and the included angle. The Law of Cosines is a versatile tool in trigonometry, applicable in various fields such as surveying, navigation, and engineering. Understanding and applying this law can greatly enhance your problem-solving abilities in geometric and real-world scenarios. This example highlights the importance of careful calculation and attention to detail in mathematical problem-solving. The Law of Cosines provides a robust method for finding missing side lengths in triangles, and this problem serves as a practical illustration of its application. By mastering this technique, you can confidently tackle similar problems and expand your understanding of trigonometry and its applications.