Problem 10 - Algebra 2

Question

WOODWORKING Latisha is to join a 6 -meter beam to a 7 -meter beam so the angle opposite the 7 -meter beam measures \(75^{\circ} .\) To what length should Latisha cut the third beam in order to form a triangular brace? Round to the nearest tenth.

Step-by-Step Solution

Answer

8.0 meters.

Step 1: Identify the Known Elements

We are given a triangle with two side lengths, 6 meters and 7 meters, and the angle between them, which is \(75^{\circ}\). This is a classic case for applying the Law of Cosines to find the third side of the triangle.

Step 2: Apply the Law of Cosines

The Law of Cosines states that for any triangle: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] where \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides. Here, \(a = 6\), \(b = 7\), and \(C = 75^{\circ}\). We'll find \(c\), the third side.

Step 3: Substitute Known Values into the Formula

Substitute \(a = 6\), \(b = 7\), and \(\cos(75^{\circ})\) into the formula: \[ c^2 = 6^2 + 7^2 - 2 \cdot 6 \cdot 7 \cdot \cos(75^{\circ}) \] Calculate the individual components.

Step 4: Calculate Each Component

First, compute the squares of the sides: \(6^2 = 36\) and \(7^2 = 49\), giving: \[ c^2 = 36 + 49 - 84 \cdot \cos(75^{\circ}) \] Compute \(84 \cdot \cos(75^{\circ})\).

Step 5: Approximate \(\cos(75^{\circ})\) and Finish Calculations

Approximate \(\cos(75^{\circ})\) using a calculator or cosine table: \(\cos(75^{\circ}) \approx 0.2588\). Substitute back: \[ c^2 = 36 + 49 - 84 \times 0.2588 \approx 36 + 49 - 21.7392 \] Thus, \[ c^2 \approx 63.2608 \] Now, find \(c\) by taking the square root.

Step 6: Calculate the Square Root

Calculate \(c\) : \[ c = \sqrt{63.2608} \approx 7.95 \] Round the result to the nearest tenth: \(c \approx 8.0\) meters.

Key Concepts

Triangle Calculations

Triangle calculations often involve determining unknown sides or angles using known measurements. A common scenario in these calculations is when you have two sides and the angle between them, as in the case of Latisha's beams forming a triangular brace.
This situation is ideal for applying the Law of Cosines, a key principle in trigonometry used to find missing side lengths.
When performing any triangle calculation, it is crucial to use accurate measurements and conversions, especially when dealing with angles expressed in degrees.
In this exercise, the triangle consists of beams (sides) of lengths 6 meters and 7 meters, with an angle of 75 degrees between them.
To find the length of the missing side (the third beam), you apply the Law of Cosines, which allows for calculating a side given two other sides and the angle between them. By systematically substituting the known values into the formula, and calculating step-by-step, you ensure accuracy in finding the length necessary to form the desired structure with precision.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It introduces important formulas and concepts such as the Law of Cosines, which is crucial for solving many real-world problems involving triangles.
The Law of Cosines is expressed as:

• \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

where:

• \(c\) is the side opposite angle \(C\)
• \(a\) and \(b\) are the other two sides

Applying this formula allows you to connect the dots between what you have (angles and sides) and what you need to find out (another side or angle in the triangle).
In our woodworking example, using \(a=6\) meters, \(b=7\) meters, and angle \(C=75^{\circ}\), we can calculate the needed length of the third side effectively.
Trigonometry is not just about calculations; it offers tools and techniques to solve various orientation and design problems through precise angle and side analyses.

Angle Measurement

Angle measurement is fundamental in solving trigonometry problems, especially when calculating triangles. In this exercise, measuring the angle correctly is essential because it determines the accuracy of our entire calculation.
Most trigonometry formulas require angles to be measured in degrees. It's always advisable to ensure that the angle measurements are precise, as incorrect angles can lead to significant errors in calculations.
For the Law of Cosines, the angle used determines the side length opposite to that angle. In Latisha's case, the angle between the 6-meter and 7-meter beams is \(75^{\circ}\). Originally identified as a known element, this value must be used precisely within the cosine function of the formula.
The cosine of an angle, such as \(75^{\circ}\) in this scenario, might not be an easy number to compute without the help of a calculator or a cosine table. A typical value you'll get for \(\cos(75^{\circ})\) is approximately 0.2588.
If you have correctly measured and calculated the angle, you ensure that the third side of the triangle will fit perfectly at the calculated 8.0 meters, maintaining the triangle's structure.