Problem 23 - Algebra 2 and Trigonometry

Question

A walking trail is laid out in the shape of a triangle. The lengths of the three paths that make up the trail are \(2,500\) meters, \(2,000\) meters, and \(1,800\) meters. Determine, to the nearest degree, the measure of the greatest angle of the trail.

Step-by-Step Solution

Answer

The greatest angle of the trail is approximately 82 degrees.

Step 1: Identifying the Sides of the Triangle

Identify the sides of the triangle. Let the sides be named as follows: - Side A = 1800 meters - Side B = 2000 meters - Side C = 2500 meters This makes side C the longest side, which is opposite the greatest angle.

Step 2: Using the Law of Cosines

To find the angle corresponding to the longest side, use the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] where C is the angle opposite side C. In this equation, insert the values: \[ 2500^2 = 1800^2 + 2000^2 - 2(1800)(2000) \cdot \cos(C) \]

Step 3: Solving for Cosine

Calculate the squares and insert them into the equation:\[ 6250000 = 3240000 + 4000000 - 7200000 \cdot \cos(C) \]Simplify the equation:\[ 6250000 = 7240000 - 7200000 \cdot \cos(C) \]

Step 4: Isolating Cosine(C)

Rearrange the equation to solve for \( \cos(C) \):\[ 7200000 \cdot \cos(C) = 7240000 - 6250000 \]\[ 7200000 \cdot \cos(C) = 990000 \]\[ \cos(C) = \frac{990000}{7200000} \]\[ \cos(C) = \frac{11}{80} \approx 0.1375 \]

Step 5: Finding the Angle C

Calculate the angle by taking the inverse cosine of the result. Use a calculator:\[ C = \cos^{-1}(0.1375) \]Calculate the angle to find that:\[ C \approx 82^\circ \]

Key Concepts

Understanding Triangle Sides

A triangle is a shape that is defined by its three sides. In most triangular geometry problems, the sides are often given specific names or labels. This helps us keep track of which sides correspond to which angles.

For our given problem, the sides are presented as follows:

• Side A = 1800 meters
• Side B = 2000 meters
• Side C = 2500 meters

These sides are labeled in increasing order, which makes it easier to identify Side C as the longest side.

Why is it important to identify the longest side? It's crucial because it helps us determine which angle is the largest. Typically, the largest angle in a triangle is opposite the longest side. In this case, we focus on calculating this angle, as it's opposite Side C. It is fundamental to lay down these foundations before delving into trigonometric formulas like the Law of Cosines.

Steps in Angle Calculation

Calculating the angle opposite the longest side in a triangle involves several steps. We use trigonometry concepts like the Law of Cosines. This special formula allows us to calculate an angle using the lengths of all three sides. Here’s how it works:

The Law of Cosines formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] This equation helps find angle C, which is opposite side C. Here, 'a' and 'b' represent the lengths of the other two sides. Substituting the side lengths into the formula gives us a way to calculate this specific angle.

In our example, plug in the numbers:\[ 2500^2 = 1800^2 + 2000^2 - 2(1800)(2000) \cdot \cos(C) \] Following this step, calculate the squares and simplify your equation to get the values lined up properly. This simplification prepares you for the next step: isolating the cosine part of the equation.

Using Inverse Cosine

The inverse cosine function, often denoted as \(\cos^{-1}\), is a powerful tool that helps us determine an angle when the cosine of that angle is known. Once we have isolated \(\cos(C)\), we need to find the angle itself.

After simplifying the Law of Cosines equation, you'll end up with an equation such as:\[ \cos(C) = \frac{11}{80} \approx 0.1375 \] With \(\cos(C)\) known, we switch gears to using the inverse cosine to find angle C:\[ C = \cos^{-1}(0.1375) \] Grab your calculator to compute this value, which results in:\[ C \approx 82^\circ \] This is the measure of the greatest angle in the triangle. Using the inverse cosine is a crucial step when transitioning from the cosine of an angle to finding the angle itself. This function bridges the gap between side lengths and angles, allowing us to fully understand the triangle's dimensions.