Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ b=9, c=16, \sin A=\frac{5}{6} $$

Step-by-Step Solution

Answer

The area of \( \triangle ABC \) is 60 square units.

Step 1: Understanding the Given Data

We are given two sides of the triangle: \( b = 9 \) and \( c = 16 \). We also know \( \sin A = \frac{5}{6} \). Our goal is to find the area of the triangle \( \triangle ABC \).

Step 2: Applying the Sine Area Formula

The formula for the area of a triangle using two sides and the sine of the included angle is given by: \[ \text{Area} = \frac{1}{2} \times b \times c \times \sin A \] Substitute the given values: \[ \text{Area} = \frac{1}{2} \times 9 \times 16 \times \frac{5}{6} \]

Step 3: Simplifying the Expression

First, compute \( 9 \times 16 = 144 \). Then substitute into the equation: \[ \text{Area} = \frac{1}{2} \times 144 \times \frac{5}{6} \] Next, simplify the fraction: \( \frac{5}{6} \times \frac{144}{1} = 120 \).

Step 4: Final Calculation to Find the Area

Now, multiply \( \frac{1}{2} \times 120 = 60 \). So, the area of \( \triangle ABC \) is 60 square units.

Key Concepts

Sine Rule

The Sine Rule is a powerful concept in trigonometry, particularly useful for solving triangles where sides and angles are not all right angles. Unlike the Pythagorean Theorem that only works on right triangles, the Sine Rule can be used for any type of triangle, be it acute, obtuse, or right.

• The Sine Rule states that: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Where \(a, b, c\) are the lengths of the sides, and \(A, B, C\) are the opposite angles.
• This rule allows you to find missing side lengths or angles when you are given some sides and angles of a triangle.

In the case of finding the area of a triangle with two sides and the sine of the included angle, the Sine Rule guides the calculation of that area through its relationship with the sine of the angle. When you know two sides and an angle, the formula for area using the sine of the angle becomes an extension of the convenience provided by the Sine Rule.

Trigonometry

Trigonometry is the branch of mathematics that deals with the properties of triangles, particularly the relationships between the angles and the lengths of their sides. It's a foundation for understanding geometric properties and is used extensively in fields ranging from engineering to computer science.

• Trigonometric functions such as sine, cosine, and tangent are crucial in this branch. They relate the angles to the ratios of the sides of right-angled triangles.
• In non-right triangles, these functions help in calculating unknown sides and angles using tools like the Sine and Cosine rules.

The use of trigonometry in determining the area of a triangle shows its versatility. For example, the Sine Area Formula \( \frac{1}{2}ab\sin C \) directly uses trigonometric principles to calculate the area when two sides and the included angle are known. This highlights the importance of understanding trigonometric functions beyond the realm of right triangles.

Triangle Geometry

Triangle geometry focuses on the properties and relations of the triangle, which is a polygon with three edges and three vertices. Every triangle has specific elements such as sides, angles, and sometimes altitudes and medians, each helping describe its shape and size.

• Triangles can be classified by their side lengths (equilateral, isosceles, scalene) or by their internal angles (acute, obtuse, right).
• The relationships between different elements of a triangle are key in many geometric problems and solutions. For instance, knowing two sides and an included angle can help determine areas, which is critical in many applications from architecture to machine design.

Understanding these basic concepts of triangle geometry is essential as they serve as the foundation for more advanced concepts and applications. When calculating areas using known sides and angles, you are essentially applying these fundamental principles of triangle geometry to arrive at precise results.