Problem 45 - 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 96 degrees.

Step 1: Determine if the Triangle is Valid

To verify if a triangle can be formed, check if the sum of the lengths of any two sides is greater than the length of the third side. 1. Check if 22 + 36 > 44 (which is 58 > 44, true). 2. Check if 22 + 44 > 36 (which is 66 > 36, true). 3. Check if 36 + 44 > 22 (which is 80 > 22, true). Since all conditions satisfy, the triangle is valid.

Step 2: Use the Law of Cosines

The Law of Cosines helps in finding the angles of a triangle when all three sides are known. It states:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]where \( c = 44 \) (the longest side, opposite to the largest angle), \( a = 22 \), \( b = 36 \), and \( C \) is the angle opposite side \( c \).

Step 3: Substitute the Values into the Law of Cosines

Plug the values into the formula:\[ 44^2 = 22^2 + 36^2 - 2 \times 22 \times 36 \times \cos(C) \]This simplifies to:\[ 1936 = 484 + 1296 - 1584 \cos(C) \]

Step 4: Solve for Cosine of the Angle

Rearrange and solve:\[ 1936 = 1780 - 1584 \cos(C) \]Subtract 1780 from both sides:\[ 156 = -1584 \cos(C) \]Now solve for \( \cos(C) \):\[ \cos(C) = \frac{-156}{1584} \approx -0.0984 \]

Step 5: Calculate the Largest Angle

To find \( C \), use the inverse cosine function:\[ C = \cos^{-1}(-0.0984) \]This gives approximately 95.67 degrees. Thus, the largest angle is rounded off to 96 degrees.

Key Concepts

Law of Cosines

The Law of Cosines is a powerful tool in trigonometry, enabling us to determine unknown angles of a triangle when we are familiar with the lengths of all its sides. For this, consider the formula: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]In this formula,

• \( c \) represents the length of the side opposite to the angle you want to find. In our problem, this is the longest side, \( 44 \; \text{yd} \).
• \( a \) and \( b \) are the lengths of the other two sides, which are \( 22 \; \text{yd} \) and \( 36 \; \text{yd} \) in our example.
• \( C \) is the angle opposite side \( c \).

By substituting these values into the Law of Cosines, we establish an equation linking side lengths and the cosine of the largest angle. This provides a clear pathway to solving for \( \cos(C) \) and subsequently the angle itself using inverse trigonometric functions.

Triangle Inequality

Before solving your triangle problem with formulas, it's crucial to ensure it's a real triangle. That's where the Triangle Inequality Theorem comes in. It simply states:

• The sum of the lengths of any two sides must be greater than the length of the third side.

Here's how you check it:- For sides \( 22 \; \text{yd} \) and \( 36 \; \text{yd} \), their sum \( 58 \; \text{yd} \), exceeds \( 44 \; \text{yd} \).- Sides \( 22 \; \text{yd} \) and \( 44 \; \text{yd} \) also meet the condition with a sum of \( 66 \; \text{yd} \), which is greater than \( 36 \; \text{yd} \).- The same condition holds for sides \( 36 \; \text{yd} \) and \( 44 \; \text{yd} \), adding to \( 80 \; \text{yd} \).This ensures that the given dimensions can indeed form a valid triangle, permitting the use of trigonometric rules such as the Law of Cosines.

Largest Angle Calculation

Determining the largest angle in a triangle involves applying the Law of Cosines and inverse trigonometric functions. After substituting side lengths into the Law of Cosines equation, you isolate \( \cos(C) \):\[ \cos(C) = \frac{-156}{1584} \approx -0.0984 \]With \( \cos(C) \) known, finding the angle \( C \) is a matter of using an inverse cosine function. This step is crucial:- Apply \( \cos^{-1} \) to \(-0.0984\) to determine \( C \).- In our problem, this calculates approximately \( 95.67\) degrees.The angle \( C \) is the largest, located opposite the longest side. Often, such calculations are rounded for simplicity, leading us here to round off to \( 96 \) degrees. Understanding these steps ensures clarity in tackling problems regarding angle determination in triangles.