Problem 47 - Algebra and Trigonometry

Question

A triangular field has sides of lengths 22, 36, and 44 yd. Find the largest angle.

Step-by-Step Solution

Answer

The largest angle is approximately 95.65°.

Step 1: Initial Analysis

To find the largest angle in a triangle given the sides, we can use the law of cosines. This theorem relates the lengths of the sides of the triangle with the cosine of one of its angles.

Step 2: Apply Law of Cosines

According to the law of cosines, for any triangle with sides of lengths \(a, b, c\) and opposite angles of \(A, B, C\), we have \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\). To find the largest angle, assume the longest side, 44 yd, is opposite the largest angle. So we'll calculate angle \(C\).

Step 3: Order the Triangle Sides

Identify the sides as \(a = 22\) yd, \(b = 36\) yd, and \(c = 44\) yd. Since \(44\) is the largest side, it is opposite the largest angle in the triangle, which is \(C\).

Step 4: Calculate Cosine of Angle C

Using the law of cosines: \( \cos C = \frac{a^2 + b^2 - c^2}{2ab}\). Substituting the values we get: \(\cos C = \frac{22^2 + 36^2 - 44^2}{2 \times 22 \times 36}\).

Step 5: Simplify and Compute

Calculate each part of the equation: \(22^2 = 484\), \(36^2 = 1296\), \(44^2 = 1936\). Plug these into the equation: \(\cos C = \frac{484 + 1296 - 1936}{2 \times 22 \times 36} = \frac{-156}{1584}\).

Step 6: Find Angle C

Simplify \(\frac{-156}{1584} \) which simplifies to \(-0.09848\). Using the inverse cosine function, \(C = \cos^{-1}(-0.09848)\). Calculate this using a calculator to get \(C \approx 95.65^{\circ}\).

Step 7: Verify Result

Re-examine calculations to ensure values were substituted correctly and the operations performed accurately. Confirm that angle \(C\) obtained is indeed the largest angle as it is calculated opposite the largest side.

Key Concepts

Triangular Field

A triangular field is a three-sided plot of land, shaped like a triangle. In geometry, a triangle has three sides and three angles. Each angle is formed by the intersection of two sides. When determining characteristics such as angles within a triangular field, we rely on geometric theorems like the law of cosines.

When examining a triangular field with given side lengths, it's crucial to understand how these dimensions impact the angles within the triangle. Besides theoretical exercises, this has practical applications, such as in land surveying where we need to measure angles and distances to define property boundaries.

• The sides of the field can be referred to by standard geometric notation: the side opposite angle A is a, opposite angle B is b, and opposite angle C is c.
• Knowing the side lengths is essential for calculating angles using trigonometric laws.

Understanding the triangular field's dimensions is the first step in calculating its angles.

Largest Angle

The largest angle in a triangle is always opposite the longest side. This is a key concept in triangles, where sides and angles have particular relationships;
understanding these relationships is crucial for solving geometric problems. The longest side impacts which angle is the largest, according to the triangle inequality principle.

When given the side lengths of a triangle, you can quickly determine that the side with maximum length is opposite the triangle's largest angle. This principle aids in predicting the characteristics of the triangle before performing any calculations.

• In our exercise, side length 44 yd is the longest, so it is opposite the largest angle, which we'll call angle C.
• This relationship helps us apply the law of cosines effectively to find the specific measure of the largest angle.

Recognizing this relationship eliminates guesswork and directs our calculations efficiently.

Side Lengths

The side lengths of a triangle are fundamental in determining the measurements of its angles. By understanding and calculating using the law of cosines, these side lengths can reveal much about the triangle’s shape and size.

In a triangle with side lengths labelled as a, b, and c, the angle opposite side c (in this case, the longest side) can be found using this law. It connects the geometry of the triangle with trigonometric principles, allowing for precise angle calculation.

• Using the steps in our exercise: Set sides as
  • a = 22 yd
  • b = 36 yd
  • c = 44 yd
• The relationship between these lengths helps in utilizing the formula for the law of cosines.
• This method not only confirms the triangle’s classification but also checks the accuracy of angle computation.

The side lengths play a crucial role and ensure that angle determination is based on solid principles and accurate measures.

Inverse Cosine

Inverse cosine, represented as \(\cos^{-1}\), is a function that helps find the angle measure when given the cosine value of that angle. It's a crucial tool for finalizing angle calculations once the cosine of the angle is known.

The inverse cosine function is the reverse operation of the cosine function. By using this operation on a known cosine value, you find the angle measure in degrees or radians, depending on your calculation settings.

• In our exercise: After using the law of cosines, the calculated \(\cos C = -0.09848\).
• To find angle C, apply the inverse cosine function: \(C = \cos^{-1}(-0.09848)\).
• This computes to approximately 95.65°, indicating the largest angle in the triangle.

The inverse cosine function finalizes the trigonometric calculations, transforming a derived ratio into a practical measurement of an angle.