Find The Third Side Of A Triangle Using The Law Of Cosines

Jul 10, 2025 by ADMIN 59 views

Determining the Length of the Third Side of a Triangle Using the Law of Cosines

Introduction

In the realm of geometry, triangles hold a fundamental position, serving as the building blocks for numerous shapes and structures. Among the various properties of triangles, the relationships between their sides and angles are particularly intriguing. One such relationship is encapsulated in the Law of Cosines, a powerful tool that allows us to determine the length of a triangle's side when we know the lengths of the other two sides and the angle between them. This article delves into the application of the Law of Cosines to solve a specific problem: finding the length of the third side of a triangle given two sides of length 3 and 4, and an included angle of 60 degrees.

Understanding the Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem, which applies only to right triangles. The Law of Cosines, on the other hand, is applicable to all types of triangles, whether they are acute, obtuse, or right-angled. The law states that for any triangle with sides of lengths a, b, and c, and angles A, B, and C opposite those sides, the following equation holds:

c² = a² + b² - 2ab cos(C)

Where:

c is the length of the side opposite angle C.
a and b are the lengths of the other two sides.
C is the angle between sides a and b.

This equation is the cornerstone of our solution. It elegantly connects the side lengths of a triangle to the cosine of one of its angles. By knowing three of these quantities (two sides and an included angle), we can use the Law of Cosines to solve for the fourth (the remaining side).

Applying the Law of Cosines to Our Problem

In our specific problem, we are given two sides of a triangle, let's call them a = 3 and b = 4, and the angle between them, C = 60°. Our goal is to find the length of the third side, c. To do this, we simply plug the given values into the Law of Cosines formula:

c² = 3² + 4² - 2 * 3 * 4 * cos(60°)

Now, we need to evaluate the cosine of 60 degrees. Recall from trigonometry that cos(60°) = 1/2. Substituting this value into the equation, we get:

c² = 9 + 16 - 2 * 3 * 4 * (1/2)

Simplifying the expression:

c² = 25 - 12
c² = 13

To find c, we take the square root of both sides:

c = √13

Therefore, the length of the third side of the triangle is √13. This result aligns with option C in the given choices.

Detailed Step-by-Step Solution

Let's break down the solution into a more detailed, step-by-step approach to ensure clarity and understanding:

Identify the knowns: We are given two sides of the triangle, a = 3 and b = 4, and the included angle, C = 60°.
Identify the unknown: We need to find the length of the third side, c.
Write down the Law of Cosines formula:
```
c² = a² + b² - 2ab cos(C)
```
Substitute the known values into the formula:
```
c² = 3² + 4² - 2 * 3 * 4 * cos(60°)
```
Evaluate the cosine of 60 degrees:
```
cos(60°) = 1/2
```
Substitute the value of cos(60°) into the equation:
```
c² = 3² + 4² - 2 * 3 * 4 * (1/2)
```

Simplify the equation:

c² = 9 + 16 - 12
c² = 25 - 12
c² = 13

Take the square root of both sides to solve for c:
```
c = √13
```

Thus, the length of the third side of the triangle is √13 units.

Alternative Approaches and Considerations

While the Law of Cosines provides a direct and efficient solution, let's briefly consider other potential approaches and some important considerations:

Attempting to use the Law of Sines: The Law of Sines could be used if we knew an angle and its opposite side. However, in this case, we only know one angle and the two sides adjacent to it. Therefore, the Law of Sines alone cannot directly solve for the third side.
Constructing an altitude: We could try to drop an altitude from one vertex to the opposite side, creating two right triangles. This would allow us to use trigonometric ratios and the Pythagorean theorem. However, this method involves more steps and calculations compared to the Law of Cosines.
Understanding the triangle inequality theorem: It's crucial to verify that the calculated side length is consistent with the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. In our case, 3 + 4 > √13, 3 + √13 > 4, and 4 + √13 > 3, so the triangle inequality holds.

Common Mistakes to Avoid

When applying the Law of Cosines, it's essential to avoid common pitfalls that can lead to incorrect answers. Here are some mistakes to watch out for:

Incorrectly substituting values: Ensure that the values for a, b, and C are correctly substituted into the formula. Pay close attention to which angle is opposite which side.
Forgetting the order of operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying the equation. Calculate the square terms and the cosine term before performing addition and subtraction.
Using the wrong trigonometric function: The Law of Cosines involves the cosine function. Avoid mistakenly using sine or tangent.
Not taking the square root: After calculating c², remember to take the square root to find the value of c.
Ignoring the triangle inequality: Always check if the calculated side length satisfies the triangle inequality theorem.

Conclusion

In conclusion, the Law of Cosines provides a powerful and direct method for determining the length of a triangle's third side when two sides and the included angle are known. By carefully applying the formula and avoiding common mistakes, we can efficiently solve such problems. In this specific case, we successfully found the length of the third side to be √13. This exercise highlights the importance of understanding and applying fundamental trigonometric principles in geometry. Mastering the Law of Cosines not only allows us to solve specific problems but also deepens our appreciation for the intricate relationships within triangles and their role in broader mathematical contexts. The ability to apply this law effectively is a valuable asset in various fields, including engineering, physics, and computer graphics, where triangle calculations are frequently encountered. Therefore, a solid grasp of the Law of Cosines is an investment in mathematical proficiency and problem-solving skills.