Determining Sides of a Triangle with Given Angles Using Algebra and Trigonometry

Given that a triangle has sides as three consecutive natural numbers, and its largest angle is exactly twice the smallest, we explore the mathematical methods to determine the sides of the triangle. This process involves the application of trigonometric identities, the Law of Cosines, and basic algebraic manipulation.

Introduction

Triangles are a fundamental concept in geometry, and understanding their properties has countless real-world applications. This article focuses on a specific type of triangle where the sides are consecutive natural numbers and the largest angle is twice the smallest angle. We will employ trigonometric and algebraic methods to uncover the sides of this unique triangle.

Step-by-Step Solution

Step 1: Defining Variables and Angles

Let the sides of the triangle be denoted as x-1, x, and x 1, where x is a natural number. θ will be the smallest angle, making 2θ the largest angle, and the third angle is π - 3θ.

Step 2: Application of Trigonometric Rules

Using the sine rule, we have:

[frac{sin theta}{x-1} frac{sin 2theta}{x 1} frac{sin(pi-3theta)}{x}]

The sine of double angle can be expressed as:

[sin 2theta 2sinthetacostheta]

Thus, we can write:

[frac{2sinthetacostheta}{sintheta} frac{x 1}{x-1} Rightarrow costheta frac{x 1}{2x-1}]

Next, we find the cosine of the largest angle:

[cos2theta left(frac{x 1}{2x-1}right)^2]

Furthermore, we know:

[sin(pi - 3theta) sin 3theta 3sintheta - 4sin^3theta]

Therefore:

[frac{sin(pi-3theta)}{x} frac{x-1}{x 1}Rightarrow 3sintheta - 4sin^3theta frac{x-1}{x 1}]

Rearranging, we get:

[3sintheta - 4sin^3theta - frac{x-1}{x 1} 0]

Now, we substitute the identity for cos^2theta into the above equation:

[sin^2theta 1 - cos^2theta 1 - left(frac{x 1}{2x-1}right)^2]

Equating the two expressions for sintheta, we derive:

[2x^2 - 3x - 2x 3 4x^2 - 8x - 4]

Solving the quadratic equation:

[x^2 - 5x 0 Rightarrow x 5 text{ or } x 0]

Since x is a natural number, we discard x 0 and obtain:

[x 5]

Consequently, the sides of the triangle are 4, 5, 6.

Conclusion

The sides of the triangle that satisfy the given conditions are 4, 5, 6. This solution showcases the intricate relationship between the angles and sides of a triangle, demonstrating the application of trigonometric identities and algebraic methods in solving geometric problems.