Finding The Smallest Angle In A Triangle Using The Law Of Cosines

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Finding the angles of a triangle when only the side lengths are known is a common problem in trigonometry. This article focuses on a specific problem: determining the smallest angle in a triangle with sides of 11, 13, and 19 units. We'll walk through the process step-by-step, using the Law of Cosines, and provide a detailed explanation to ensure clarity.

Understanding the Problem

We are given a triangle with sides a = 11, b = 13, and c = 19. Our goal is to find the measure of the smallest angle, which we'll denote as ∠J. The key concept here is that the smallest angle is always opposite the shortest side. In this case, the shortest side is 11, so ∠J is the angle opposite this side.

The Law of Cosines: A Powerful Tool

The Law of Cosines is a fundamental formula in trigonometry that relates the sides and angles of a triangle. It's particularly useful when we know all three sides (SSS) and want to find an angle. The Law of Cosines has three forms, but the one we'll use is:

  • a² = b² + c² - 2bc * cos(A)

Where:

  • a, b, and c are the lengths of the sides of the triangle.
  • A is the angle opposite side a.

In our case, we want to find ∠J, which is opposite the side with length 11. So, we can rewrite the formula as:

  • 11² = 13² + 19² - 2 * 13 * 19 * cos(J)

This equation allows us to solve for cos(J), and then we can use the inverse cosine function (arccos or cos⁻¹) to find the angle J.

Step-by-Step Solution

Let's break down the calculation step-by-step:

  1. Substitute the values:

    • 11² = 13² + 19² - 2 * 13 * 19 * cos(J)
    • 121 = 169 + 361 - 494 * cos(J)

  2. Simplify the equation:

    • 121 = 530 - 494 * cos(J)

  3. Isolate the term with cos(J):

    • 494 * cos(J) = 530 - 121
    • 494 * cos(J) = 409

  4. Solve for cos(J):

    • cos(J) = 409 / 494
    • cos(J) ≈ 0.8279

  5. Find the angle J using the inverse cosine function:

    • J = arccos(0.8279)
    • J ≈ 34.11 degrees

  6. Round to the nearest whole degree:

    • J ≈ 34 degrees

Therefore, the measure of the smallest angle, ∠J, is approximately 34 degrees.

Choosing the Correct Answer

Looking at the provided options:

  • A. 30°
  • B. 34°
  • C. 42°
  • D. 47°

The correct answer is B. 34°. This confirms our calculation using the Law of Cosines.

Importance of the Law of Cosines

The Law of Cosines is a vital tool in trigonometry and has numerous applications in various fields, including surveying, navigation, and engineering. It allows us to solve triangles when we have Side-Side-Side (SSS) or Side-Angle-Side (SAS) information. Understanding and applying the Law of Cosines is crucial for solving many geometric problems.

Real-World Applications

Consider a surveyor trying to determine the distance between two points on opposite sides of a lake. They can measure the distances from a third point to each of the two points and the angle between these lines of sight. Using the Law of Cosines, they can then calculate the distance across the lake without directly measuring it. This is just one example of how the Law of Cosines simplifies real-world problems.

Key Concepts Recap

To solidify your understanding, let's recap the key concepts:

  • The smallest angle in a triangle is opposite the shortest side.
  • The Law of Cosines is used to find angles when all three sides are known.
  • The formula a² = b² + c² - 2bc * cos(A) is the standard form of the Law of Cosines.
  • The inverse cosine function (arccos or cos⁻¹) is used to find the angle from its cosine value.

Common Mistakes to Avoid

When using the Law of Cosines, there are a few common mistakes to be aware of:

  1. Incorrectly identifying the opposite angle: Make sure you match the side length with the correct opposite angle in the formula.
  2. Algebraic errors: Double-check your calculations, especially when simplifying and isolating cos(A).
  3. Forgetting the inverse cosine function: Remember to use arccos (or cos⁻¹) to find the angle after calculating cos(A).
  4. Rounding errors: If you round intermediate values, it can affect the final answer. It's best to keep as many decimal places as possible until the final step.

Avoiding these mistakes will help you apply the Law of Cosines accurately and efficiently.

Practice Problems

To further enhance your understanding, try solving similar problems. For example:

  1. Find the measure of the largest angle in a triangle with sides 7, 9, and 12.
  2. A triangle has sides of lengths 5, 8, and 10. Find all three angles.

Working through these problems will solidify your understanding of the Law of Cosines and its applications.

Conclusion

In conclusion, finding the smallest angle in a triangle when given the side lengths involves using the Law of Cosines. By understanding the relationship between sides and angles and following a step-by-step approach, we can accurately determine the angle's measure. In this specific case, the smallest angle in the triangle with sides 11, 13, and 19 is approximately 34 degrees. The Law of Cosines is a versatile tool with real-world applications, making it an essential concept in trigonometry. Remember to practice and apply the concepts to various problems to master this technique.