Finding The Third Side Of A Triangle Law Of Cosines Example

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In the realm of geometry, triangles hold a fundamental position, and understanding their properties is crucial for solving a myriad of problems. One common task involves determining the length of the third side of a triangle when two sides and the included angle are known. This is where the Law of Cosines comes into play, a powerful tool that provides a direct relationship between the sides and angles of any triangle. 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 2 and 5, with the included angle being 60 degrees.

Understanding the Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem, applicable to all triangles, not just right triangles. It states that for any triangle with sides of lengths a, b, and c, and angles A, B, and C opposite to 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 law essentially relates the square of one side of a triangle to the squares of the other two sides and the cosine of the included angle. When the included angle is a right angle (90 degrees), the cosine term becomes zero, and the equation reduces to the Pythagorean theorem (c² = a² + b²).

Applying the Law of Cosines to the Problem

In our specific problem, we are given two sides of a triangle with lengths 2 and 5, and the angle between them is 60 degrees. Let's denote the sides as a = 2, b = 5, and the angle between them as C = 60 degrees. We want to find the length of the third side, which we'll call c.

Using the Law of Cosines formula:

c² = a² + b² - 2ab cos(C)

Substitute the given values:

c² = 2² + 5² - 2 * 2 * 5 * cos(60°)

Now, we know that cos(60°) = 1/2. Plugging this in:

c² = 4 + 25 - 2 * 2 * 5 * (1/2)

c² = 29 - 10

c² = 19

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

c = √19

Therefore, the length of the third side of the triangle is √19.

Detailed Calculation and Explanation

To further clarify the solution, let's break down the calculation step by step:

  1. Identify the given values:
    • a = 2
    • b = 5
    • C = 60°
  2. Write down the Law of Cosines formula:
    • c² = a² + b² - 2ab cos(C)
  3. Substitute the values into the formula:
    • c² = (2)² + (5)² - 2(2)(5)cos(60°)
  4. Calculate the squares:
    • c² = 4 + 25 - 2(2)(5)cos(60°)
  5. Evaluate the cosine:
    • cos(60°) = 1/2
  6. Substitute the cosine value:
    • c² = 4 + 25 - 2(2)(5)(1/2)
  7. Perform the multiplication:
    • c² = 4 + 25 - 10
  8. Add and subtract:
    • c² = 29 - 10
    • c² = 19
  9. Take the square root:
    • c = √19

This step-by-step approach provides a clear understanding of how the Law of Cosines is applied to solve for the unknown side. Each step is logically connected, making it easier to follow the reasoning and ensure accuracy.

Significance of the Law of Cosines

The Law of Cosines is a fundamental concept in trigonometry and geometry. Its significance lies in its ability to solve triangles when given specific information. Here's why it's so important:

  1. Solving Non-Right Triangles: Unlike the basic trigonometric ratios (sine, cosine, tangent) which are primarily used for right triangles, the Law of Cosines can be applied to any triangle, regardless of its angles. This makes it a versatile tool for solving a wider range of problems.
  2. Side-Angle-Side (SAS) and Side-Side-Side (SSS) Cases: The Law of Cosines is particularly useful in two common scenarios:
    • SAS: When two sides and the included angle are known, the Law of Cosines can be used to find the third side.
    • SSS: When all three sides are known, the Law of Cosines can be rearranged to find any of the angles.
  3. Applications in Real-World Problems: The Law of Cosines has numerous applications in various fields, including:
    • Navigation: Determining distances and bearings in navigation.
    • Engineering: Calculating forces and stresses in structures.
    • Surveying: Measuring distances and angles in land surveying.
    • Physics: Solving problems involving vectors and forces.

Its ability to handle non-right triangles and its applicability in SAS and SSS cases make it an indispensable tool in mathematics and its applications.

Alternative Methods and Considerations

While the Law of Cosines is the most direct method for solving this particular problem, it's worth noting that there might be alternative approaches, though they might be more complex or less efficient.

  1. Law of Sines: The Law of Sines could be used in conjunction with the Law of Cosines. However, it would require an extra step of finding an angle first using the Law of Cosines before applying the Law of Sines to find the third side. This method is generally less straightforward in this case.
  2. Geometric Construction and Trigonometry: One could attempt to construct the triangle and use other trigonometric relationships. This approach might involve dropping a perpendicular from one vertex to the opposite side, creating right triangles. However, this method can be more complex and prone to errors.

When choosing a method, it's important to consider the given information and the desired outcome. The Law of Cosines is the most efficient and direct method when two sides and the included angle are known.

Common Mistakes and How to Avoid Them

When applying the Law of Cosines, there are some common mistakes that students often make. Being aware of these pitfalls can help ensure accurate solutions.

  1. Incorrectly Substituting Values: A common mistake is substituting the values into the formula incorrectly. Make sure that the sides and angles are matched correctly. Remember that 'c' is the side opposite angle 'C'.
  2. Forgetting the Order of Operations: Follow the order of operations (PEMDAS/BODMAS) when evaluating the expression. Calculate the squares first, then the cosine, then the multiplication, and finally the addition and subtraction.
  3. Incorrectly Evaluating Cosine: Ensure you are using the correct value for the cosine of the angle. For common angles like 60 degrees, remember that cos(60°) = 1/2.
  4. Not Taking the Square Root: Remember that the Law of Cosines gives you c², so you need to take the square root to find the length of side c.
  5. Rounding Errors: Avoid rounding intermediate values, as this can lead to inaccuracies in the final answer. Only round the final answer to the desired level of precision.

By being mindful of these common mistakes, you can improve your accuracy and confidence when using the Law of Cosines.

Conclusion

In conclusion, the Law of Cosines provides a powerful and direct method for finding the length of the third side of a triangle when two sides and the included angle are known. In the given problem, with sides of length 2 and 5 and an included angle of 60 degrees, the length of the third side is found to be √19. The Law of Cosines is a fundamental concept in trigonometry and has wide-ranging applications in various fields. Understanding its principles and proper application is essential for solving geometric problems and real-world scenarios involving triangles.

By mastering the Law of Cosines and practicing its application, you can confidently tackle a variety of problems involving triangles and their properties. This foundational knowledge is invaluable for further studies in mathematics and related disciplines.