Problem 32 - Algebra and Trigonometry

Question

Show that, given the three angles \(A, B, C\) of a triangle and one side, say \(a,\) the area of the triangle is $$ \text { area }=\frac{a^{2} \sin B \sin C}{2 \sin A} $$

Step-by-Step Solution

Answer

The area of the triangle is \( \frac{a^{2} \, \sin B \, \sin C}{2 \, \sin A} \).

Step 1: Identify the Problem

We are given a triangle with sides labeled as \( a, b, c \) opposite to angles \( A, B, C \) respectively, and the task is to express the area of the triangle using one side and the sine of its angles.

Step 2: Use the Formula for Area of a Triangle

Recall that the area \( \Delta \) of a triangle with sides \( a, b, \) and angle \( C \) between them can be given by \( \Delta = \frac{1}{2} ab \sin C \).

Step 3: Apply the Law of Sines

The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We can solve this equation for \( b \) in terms of \( a, \sin A, \text{ and } \sin B \). Hence, \( b = \frac{a \sin B}{\sin A} \).

Step 4: Substitute Value for \( b \) into the Area Formula

Replace \( b \) in the area formula with \( \frac{a \sin B}{\sin A} \), leading to \( \Delta = \frac{1}{2} \left( a \cdot \frac{a \sin B}{\sin A} \right) \sin C \).

Step 5: Simplify the Expression

Simplify the expression for the area: \( \Delta = \frac{1}{2} \cdot \frac{a^2 \sin B \sin C}{\sin A} \).

Step 6: Final Verification

The final expression matches the given formula: \( \text{area} = \frac{a^{2} \, \sin B \, \sin C}{2 \, \sin A} \), verifying that the solution steps lead to the correct area expression.

Key Concepts

Law of Sines

The Law of Sines is a fundamental theorem in trigonometry that helps you solve triangles. It relates the sides of a triangle to the sines of its angles. Essentially, it states that for any triangle with angles \( A, B, \) and \( C \) and sides \( a, b, \) and \( c \) opposite these angles, the following is true:

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

This relationship allows us to find unknown side lengths or angles when some of them are known. In our exercise, this law is used to express one side of the triangle in terms of another side and the sines of the angles. Specifically, it lets you calculate \( b \) in terms of \( a \), \( \sin B \), and \( \sin A \) by rearranging the law to \( b = \frac{a \sin B}{\sin A} \). This substitution is crucial for deriving the expression for the area of the triangle.

Trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles, particularly right triangles, and the relationships between their angles and sides. The sine function, which is a key player in trigonometry, measures the ratio of the opposite side to the hypotenuse in a right-angled triangle:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the context of any triangle, trigonometry provides tools like the Law of Sines, which allow us to explore and calculate areas, side lengths, and angles. In this exercise, trigonometry helps us understand how the angle sine values relate to the sides, enabling us to find the area using these relationships.

Triangle Geometry

Triangles are one of the most basic geometric shapes and have unique properties. A triangle has three sides, three angles, and the sum of its interior angles is always \( 180^{\circ} \). In triangle geometry, different formulas can be used to evaluate properties such as side lengths, angles, and area.

Finding the area of a triangle can depend on what information is available. If we know two sides and the included angle, we can use the formula:

• \( \Delta = \frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{angle between them}) \)

This formula effectively uses one of the fundamental concepts of triangle geometry: the relationship between the components of a triangle. In the original exercise, even if only one side and angles are given, we can find the area using the properties of triangles and trigonometry. These techniques reduce complex geometry problems to more manageable calculations.