Finding Side Length C In Triangle AABC A Law Of Cosines Application

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In this article, we delve into a fascinating problem involving triangle geometry. We're given a triangle AABC, where we know the measure of angle A (m∠A = 15°), the length of side 'a' (a = 10), and the length of side 'b' (b = 11). Our mission is to find the length of side 'c' to the nearest tenth. This type of problem commonly appears in trigonometry and can be elegantly solved using the Law of Cosines. The Law of Cosines is a fundamental concept in trigonometry that relates the sides and angles of any triangle, not just right triangles. It's a versatile tool that allows us to solve for unknown side lengths or angles when we have sufficient information about the other parts of the triangle. In this particular case, we have two sides and an included angle, which perfectly sets the stage for applying the Law of Cosines. The solution involves careful application of the formula, algebraic manipulation, and a bit of numerical computation. Understanding the Law of Cosines and its applications is crucial for anyone studying trigonometry, geometry, or related fields. It's a building block for solving more complex problems in surveying, navigation, and engineering. Before we dive into the solution, let's briefly review the Law of Cosines and understand its significance. This will provide a solid foundation for understanding the steps involved in solving for side 'c'. We will also discuss why this approach is the most suitable one for this type of problem, highlighting its advantages over other trigonometric methods. So, let's embark on this journey of discovery and unravel the mystery of side 'c' in triangle AABC.

Understanding the Law of Cosines

The Law of Cosines is a powerful extension of the Pythagorean theorem, applicable to all triangles, whether they are right-angled or not. It provides a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. The formula for the Law of Cosines is expressed as follows:

  • c² = a² + b² - 2ab * cos(C)

Where:

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

The Law of Cosines is a versatile tool that can be used in various scenarios. It's particularly useful when we have the following information:

  • Two sides and the included angle (SAS): This is the case in our problem, where we know sides 'a' and 'b' and the angle A between them.
  • Three sides (SSS): If we know the lengths of all three sides, we can use the Law of Cosines to find any of the angles.

The Law of Cosines is a cornerstone of trigonometry, providing a way to solve triangles that are not right-angled. It's essential for various applications in surveying, navigation, engineering, and physics. Understanding its derivation and applications is crucial for anyone working with triangles and geometric problems. The Law of Cosines essentially bridges the gap between the side lengths and angles in any triangle, making it a fundamental tool in trigonometry. It allows us to solve for unknown side lengths or angles when we have sufficient information about the other parts of the triangle. In our case, we have two sides and the included angle, which makes the Law of Cosines the perfect choice for finding the length of side 'c'. The formula elegantly captures the relationship between the sides and the cosine of the angle, allowing us to directly calculate the unknown side length. Let's now see how we can apply this powerful law to solve our specific problem.

Applying the Law of Cosines to Solve for 'c'

In our problem, we are given the following information:

  • m∠A = 15°
  • a = 10
  • b = 11

We want to find the length of side 'c'. To apply the Law of Cosines, we need to rearrange the formula to solve for 'c'. Since we have angle A, we'll use the form of the Law of Cosines that involves angle A:

  • a² = b² + c² - 2bc * cos(A)

However, we want to solve for 'c', so we need to rearrange the formula to have 'c' on one side. But in our case, we know angle A, and sides a and b, so we should use the form:

  • c² = a² + b² - 2ab * cos(C)

Wait! We don't have angle C. We actually need to adapt the formula to use angle A:

  • a² = b² + c² - 2bc * cos(A)

Now, we can plug in the given values:

  • 10² = 11² + c² - 2 * 11 * c * cos(15°)

This gives us a quadratic equation in terms of 'c'. Let's simplify and rearrange the equation:

  • 100 = 121 + c² - 22c * cos(15°)

  • 0 = c² - 22 * cos(15°) * c + 21

Now, we have a quadratic equation in the form of:

  • ax² + bx + c = 0

Where:

  • a = 1
  • b = -22 * cos(15°)
  • c = 21

We can solve this quadratic equation using the quadratic formula:

  • x = (-b ± √(b² - 4ac)) / 2a

In our case, 'x' represents 'c'. Let's plug in the values and calculate the solutions for 'c'. This step involves careful arithmetic and the use of a calculator to find the cosine of 15 degrees. We'll obtain two possible values for 'c', and we'll need to consider both to see if they are valid solutions for the side length of a triangle. The Law of Cosines provides us with a powerful tool to set up the equation, and the quadratic formula allows us to solve for the unknown side length. The next step is to carefully perform the calculations and interpret the results in the context of the problem.

Solving the Quadratic Equation

Now, let's plug the values into the quadratic formula:

  • c = (-(-22 * cos(15°)) ± √((-22 * cos(15°))² - 4 * 1 * 21)) / (2 * 1)

First, we need to calculate cos(15°). Using a calculator, we find that:

  • cos(15°) ≈ 0.9659

Now, substitute this value back into the quadratic formula:

  • c = (22 * 0.9659 ± √((22 * 0.9659)² - 84)) / 2

  • c = (21.25 ± √(452.81 - 84)) / 2

  • c = (21.25 ± √368.81) / 2

  • c = (21.25 ± 19.20) / 2

This gives us two possible solutions for 'c':

  • c₁ = (21.25 + 19.20) / 2 ≈ 20.23

  • c₂ = (21.25 - 19.20) / 2 ≈ 1.03

We have obtained two possible values for the length of side 'c'. However, we need to consider the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Let's check if both solutions satisfy this condition.

For c₁ ≈ 20.23:

  • a + b = 10 + 11 = 21 > 20.23 (True)
  • a + c = 10 + 20.23 = 30.23 > 11 (True)
  • b + c = 11 + 20.23 = 31.23 > 10 (True)

For c₂ ≈ 1.03:

  • a + b = 10 + 11 = 21 > 1.03 (True)
  • a + c = 10 + 1.03 = 11.03 > 11 (False!)
  • b + c = 11 + 1.03 = 12.03 > 10 (True)

Since c₂ ≈ 1.03 violates the triangle inequality theorem (specifically, a + c is not greater than b), it is not a valid solution. Therefore, the only valid solution for 'c' is approximately 20.23. Rounding this to the nearest tenth, we get c ≈ 20.2. The process of solving the quadratic equation and checking the validity of the solutions is a crucial step in ensuring we arrive at the correct answer. The triangle inequality theorem acts as a safeguard, preventing us from accepting solutions that would not form a valid triangle.

Verifying the Solution and Final Answer

We have found two possible solutions for 'c' using the quadratic formula, approximately 20.23 and 1.03. We then applied the triangle inequality theorem to determine which solution(s) were valid. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. We found that the solution c₂ ≈ 1.03 violated this theorem, as 10 + 1.03 is not greater than 11. Therefore, we discarded this solution.

The other solution, c₁ ≈ 20.23, satisfies the triangle inequality theorem, as:

  • 10 + 11 > 20.23
  • 10 + 20.23 > 11
  • 11 + 20.23 > 10

Thus, c₁ ≈ 20.23 is a valid solution. Rounding to the nearest tenth, we get c ≈ 20.2.

Therefore, the length of side 'c' in triangle AABC, to the nearest tenth, is 20.2. This final step of verification is essential to ensure that the solution we have obtained is not only mathematically correct but also makes sense in the context of the geometric problem. By applying the triangle inequality theorem, we have confirmed that our solution is indeed a valid one. The entire process, from applying the Law of Cosines to solving the quadratic equation and verifying the solution, demonstrates a thorough understanding of trigonometric principles and problem-solving techniques. The final answer, c ≈ 20.2, is the culmination of this rigorous process.

Conclusion

In this article, we successfully found the length of side 'c' in triangle AABC using the Law of Cosines. We were given m∠A = 15°, a = 10, and b = 11. By applying the Law of Cosines, we derived a quadratic equation and solved it using the quadratic formula. We obtained two possible solutions for 'c', but after applying the triangle inequality theorem, we determined that only one solution was valid. The correct solution, rounded to the nearest tenth, is c ≈ 20.2.

This problem highlights the power and versatility of the Law of Cosines in solving triangle problems. It also demonstrates the importance of verifying solutions to ensure they are valid in the context of the problem. The Law of Cosines is a fundamental tool in trigonometry and has numerous applications in various fields, including surveying, navigation, and engineering. Mastering this concept is crucial for anyone working with triangles and geometric problems. The process we followed in this article provides a clear and systematic approach to solving similar problems. It involves careful application of the formula, algebraic manipulation, and a thorough understanding of the underlying principles. The final answer, c ≈ 20.2, is a testament to the effectiveness of the Law of Cosines and the importance of verifying solutions. We hope this article has provided a clear and comprehensive understanding of how to solve this type of problem and has reinforced the importance of the Law of Cosines in trigonometry.

Final Answer: C. 20.2