Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the area of triangle \(A B C\). $$\alpha=80.1^{\circ}, \quad a=8.0, \quad b=3.4$$

Step-by-Step Solution

Answer

The approximate area of triangle \(ABC\) is about 13.396 square units.

Step 1: Understanding the Problem

We need to approximate the area of triangle \(ABC\) using the given angle \(\alpha = 80.1^{\circ}\) and the sides \(a = 8.0\) and \(b = 3.4\). The angle \(\alpha\) is the angle opposite side \(c\) and between sides \(a\) and \(b\).

Step 2: Choosing the Formula for Area

To approximate the area of the triangle, we can use the formula: \( \text{Area} = \frac{1}{2}ab\sin(\alpha) \). This formula is applicable because we know the lengths of two sides and the angle between them.

Step 3: Calculating \(\sin(\alpha)\)

First, we calculate \(\sin(80.1^{\circ})\). Using a calculator, we find that \(\sin(80.1^{\circ}) \approx 0.985\).

Step 4: Applying the Formula

Substitute the given values into the formula: \( \text{Area} = \frac{1}{2} \times 8.0 \times 3.4 \times 0.985 \).

Step 5: Performing the Multiplication

Calculate the multiplication step by step: \( 8.0 \times 3.4 = 27.2 \), then \( 27.2 \times 0.985 \approx 26.792 \).

Step 6: Calculating the Final Area

Finally, calculate \( \frac{1}{2} \times 26.792 \approx 13.396 \). Thus, the approximate area of triangle \(ABC\) is about \(13.396\) square units.

Key Concepts

Sine rule

The Sine Rule is a fundamental concept in trigonometry, useful for solving unknowns in triangles. It relates the sides of a triangle to the sines of its angles, and it's especially handy when the triangle is not a right triangle. The rule states that in any given triangle, the ratio of the length of a side to the sine of its opposite angle is constant for each side. In formula terms:

• \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

This rule can be used to find missing sides or angles when certain other elements of the triangle are known. In the current exercise, while the Sine Rule directly isn't used to calculate the area, understanding how the sine of an angle functions helps us employ the area formula that involves sine.

Trigonometry

Trigonometry is the branch of mathematics that explores relationships within triangles. It primarily deals with the study of angles and lengths of triangles. The core functions of trigonometry—sine, cosine, and tangent—define these relationships and allow us to calculate unknown dimensions.
For example, in our task to find the area of triangle \(ABC\), one of these core functions, sine, plays a central role. The expression \( \sin(\alpha) \) helps us determine the area by using the formula:

• \( \text{Area} = \frac{1}{2}ab\sin(\alpha) \)

This formula is profound because it allows us to calculate the area of a triangle when two sides and the included angle are known, leveraging trigonometric principles. The trigonometric functions allow for straightforward calculations that would otherwise be quite complex without them.

Triangle properties

Understanding the properties of triangles is crucial, as it lays the groundwork for applying various mathematical rules and formulas. Every triangle has three sides and three angles, which add up to 180 degrees. Each angle corresponds to the side that faces it, and the relationships between these aspects determine which mathematical strategies we can use.

• In a triangle such as \(ABC\), knowing two sides and one angle—particularly the included angle—empowers us to calculate other properties, such as area, using specific formulas.
• The given exercise utilizes the concept that knowing two sides and the included angle (an angle between the two sides) allows us to find the third side or the area efficiently.

By applying these triangle properties, the formula \( \text{Area} = \frac{1}{2}ab\sin(\alpha) \) becomes applicable due to the known values of two sides and the angle between them. This property of triangles is a critical aspect of solving a wide array of geometric problems.