Problem 16 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=3.7 \mathrm{~m}, c=6.4 \mathrm{~m} \text {, and } B=33^{\circ} \text {, find } b \text {. } $$

Step-by-Step Solution

Answer

The length of side \( b \) is approximately \( 3.86 \, \text{m} \).

Step 1: Identify known values

Firstly, recognize what values are given in the problem. We know that \( a = 3.7 \, \text{m} \), \( c = 6.4 \, \text{m} \), and angle \( B = 33^{\circ} \). We are tasked with finding side \( b \).

Step 2: Use the Law of Cosines formula

Since we have two sides and the included angle, we can use the law of cosines to find the missing side. The formula is \( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \).

Step 3: Plug in the known values

Substitute the known values into the equation: \[ b^2 = (3.7)^2 + (6.4)^2 - 2 \cdot 3.7 \cdot 6.4 \cdot \cos(33^{\circ}) \]

Step 4: Calculate the expression

First, calculate each part step-by-step:1. \( 3.7^2 = 13.69 \)2. \( 6.4^2 = 40.96 \)3. Both terms combined: \( 13.69 + 40.96 = 54.65 \)Calculate \( 2 \cdot 3.7 \cdot 6.4 \cdot \cos(33^{\circ}) \) using cosine:1. \( 2 \cdot 3.7 \cdot 6.4 = 47.36 \)2. Multiply by \( \cos(33^{\circ}) \approx 0.8387 \)3. \( 47.36 \cdot 0.8387 \approx 39.72 \) (rounded to two decimal places)

Step 5: Solve for \( b^2 \) and \( b \)

Subtract the products from our previous steps: \[ 54.65 - 39.72 = 14.93 \]Thus, \( b^2 = 14.93 \). Now, take the square root to find \( b \): \[ b = \sqrt{14.93} \approx 3.86 \]

Step 6: Conclude with the value of \( b \)

The length of side \( b \) is approximately \( 3.86 \, \text{m} \).

Key Concepts

Trigonometry: Understanding the Basics

Trigonometry is a branch of mathematics that explores the relationships between the sides and angles of triangles. At its core, trigonometry is about connecting geometric problems to algebra and calculus. This involves mixing shapes and measurements with numbers and formulas. Three main functions in trigonometry—sine, cosine, and tangent—relate angles of a triangle to side lengths. These functions are the foundation of many formulas used in trigonometry today.

• Sine (sin) relates the angle to the opposite side divided by the hypotenuse.
• Cosine (cos) involves the adjacent side and the hypotenuse.
• Tangent (tan) is the ratio of the opposite side over the adjacent side.

The law of cosines, which we use in our exercise, modifies the basic structure of the cosine function specifically for non-right triangles. It allows us to find unknown sides when we know two sides and an included angle. Understanding these basics helps decipher more complex geometric problems and finds practical applications in fields like engineering and physics.

Triangle Calculations: Using the Law of Cosines

Calculating unknown measurements in triangles often involves strategic use of various mathematical laws and formulas. The Law of Cosines is particularly useful in cases where traditional right-angled trigonometry is not applicable. This makes it great for dealing with scalene triangles where all sides and angles are different.
The formula for the Law of Cosines is given as:\[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \]

• This works similarly to the Pythagorean Theorem but with a twist—the inclusion of the angle's cosine.
• We apply it when we know two sides and their included angle or three sides.

In the given problem, we had two sides: 'a' and 'c', and the included angle 'B.' By plugging in these values, we calculated the third side 'b.' This step-by-step breakdown of calculations ensures that no detail is overlooked, enhancing accuracy and understanding.

Angle-Side Relationships in Triangles

Understanding the fundamental relationships between angles and sides is critical for solving problems involving triangles. These relationships form the building blocks of many concepts in geometry and trigonometry. In any triangle, the sum of interior angles always equals 180 degrees; however, it's how these angles relate to the sides that becomes truly interesting.

• In scalene triangles, where all sides and angles differ, the largest angle will always be opposite the longest side.
• Similarly, the smallest angle is opposite the shortest side.

The Law of Cosines encapsulates this relationship between angles and sides in its formulaic structure. It allows us to find a missing side when two sides and the respective angle between them are known. This is why understanding angle-side relationships is crucial. They not only guide the setup of a problem but also inform which formulas are most applicable to find a solution. Through the Law of Cosines, we're able to maintain these relationships while solving for unknown sides, crucial for achieving accurate and logical results in geometric computations.