Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ b=7, c=8, \sin A=\frac{3}{5} $$

Step-by-Step Solution

Answer

The area of triangle ABC is 16.8 square units.

Step 1: Understanding the Given Data

The problem provides us with specific details about triangle ABC. We know side \(b = 7\), side \(c = 8\) and \(\sin A = \frac{3}{5}\). Our task is to find the area of triangle ABC using these values.

Step 2: Using the Formula for the Area of a Triangle

We can calculate the area of triangle ABC using the formula for the area of a triangle given two sides and the sine of the included angle: \[ \text{Area} = \frac{1}{2} \times b \times c \times \sin A \] Substitute the given values into this formula.

Step 3: Substituting the Values into the Formula

Replace \(b\), \(c\), and \(\sin A\) with the given values:\[ \text{Area} = \frac{1}{2} \times 7 \times 8 \times \frac{3}{5} \]

Step 4: Simplifying the Expression

Calculate the multiplication in the formula step-by-step:First, calculate \(7 \times 8 = 56\).Then, \(56 \times \frac{3}{5} = \frac{168}{5}\).Finally, divide by 2:\[ \frac{1}{2} \times \frac{168}{5} = \frac{168}{10} = 16.8 \] Thus, the area of triangle ABC is 16.8 square units.

Key Concepts

Sine Rule

The Sine Rule is a fundamental tool in trigonometry, especially when dealing with triangles. It states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles. This is crucial in scenarios where some angles or sides are unknown, and it provides a way to solve them. The formula for the Sine Rule is:

• For triangle with sides a, b, c and angles A, B, C opposite these sides respectively: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

In the exercise, although we didn't directly utilize the sine rule, it is foundational knowledge that gives us insight into the properties of triangle ABC. Specifically, the information \( \sin A = \frac{3}{5} \) directly helps in computing the area of the triangle by multiplying it with the side lengths \( b \) and \( c \). Knowing the sine rule can also help us find other unknown angles or side lengths if required.

Trigonometry

Trigonometry may sound complex, but it is essentially the study of triangles and the relationships between their sides and angles. In particular, it is useful for calculating distances and angles in various fields such as engineering, physics, and even architecture. Specifically, trigonometry is broken down into several fundamental functions: sine, cosine, and tangent. Each function provides different insights based on different angle measurements and their related side lengths. In the context of the exercise, we use the sine function, which is part of basic trigonometry, to find the area of the triangle:

• The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• We utilize \( \sin A = \frac{3}{5} \) to compute half of the area of \( \triangle ABC \).

Understanding this allows us to link angles and side lengths directly, converting our problem into simple multiplication by using the given values, making the task straightforward.

Geometry

Geometry is at the heart of understanding shapes, spaces, and how they interact. It's the branch of mathematics that deals with points, lines, surfaces, shapes, and solids. In particular, triangle geometry is a key topic because triangles serve as building blocks for constructing more complex geometrical figures. The area of a triangle is calculated using different methods depending on the provided information. When two sides and the included angle are known, the area can be calculated using:

• The formula \( \text{Area} = \frac{1}{2} \times b \times c \times \sin A \).
• This concept is rooted in both basic geometric principles and trigonometry.

This method is particularly useful for non-right-angled triangles, where alternative basic formulas like \( \frac{1}{2} \times base \times height \) are not applicable without additional height information. By understanding the principles of geometry and integrating trigonometric functions, students can solve the exercise efficiently and understand the broader application of these concepts.