Solving Trigonometric Equations: A Comprehensive Guide with Multiple Methods

Trigonometric equations are a fundamental part of advanced mathematics, particularly within calculus and physics. Solving these equations can often feel like an art, with various methods and identities at one's disposal. In this article, we will explore different techniques to solve a specific trigonometric equation, demonstrating multiple methods that lead to the same result.

Introduction to the Problem

The given problem involves solving the following equation:

Equation: [2cos^{2}x 2cos^{2}2x 2cos^{2}3x 2]

This equation requires a keen understanding of trigonometric identities and techniques. The methods we will explore include the double angle identity, substitution, and complex exponential techniques.

Solution via Double Angle Identity

Let's start by applying the double angle identity, which states that:

Double Angle Identity: [cos 2x 2cos^2 x - 1]

We can use this identity to simplify our main equation. First, let's rewrite the equation using this identity:

[2(1 cos 2x) 2(1 cos 4x) 2(1 cos 6x) 2]

Simplifying further:

[2 2cos 2x 2 2cos 4x 2 2cos 6x 2]

This reduces to:

[6 2cos 2x 2cos 4x 2cos 6x 2]

Which simplifies to:

[2cos 2x 2cos 4x 2cos 6x -4]

Dividing by 2:

[cos 2x cos 4x cos 6x -2]

This equation can be further simplified using trigonometric product-to-sum identities, but for two specific solutions:

[cos 2x 2cos^2 2x - 1 2cos^2 2x - 1 - cos 2x 2]

[4cos^2 2x - cos 2x - 1 0]

Letting (y cos 2x), we get a quadratic equation:

[4y^2 - y - 1 0]

Solving this quadratic equation using the quadratic formula:

[y frac{-b pm sqrt{b^2 - 4ac}}{2a}]

[y frac{1 pm sqrt{1 16}}{8}]

[y frac{1 pm sqrt{17}}{8}]

However, for simplicity, examining the roots in the unit circle, we get specific angles:

[cos 2x 0, frac{1}{2} , -1]

These solutions translate to:

[x pmfrac{pi}{4} npi, pmfrac{pi}{6} npi, pmfrac{pi}{2} npi]

Solution via Complex Exponential Form

Another powerful method is using the complex exponential form of cosine:

Complex Exponential Form: [cos x frac{e^{ix} e^{-ix}}{2}]

Cubing this form, we get:

Triple Angle Identity: [cos 3x 4cos^3 x - 3cos x]

Substituting these identities into our original equation, we get the polynomial:

[cos^2 x 8cos^4 x - 10cos^2 x - 3 0]

This is a quadratic in (cos^2 x), which can be factored to:

[cos^2 x 2cos^2 x - 1 - 4cos^2 x - 3 0]

The solutions to this quadratic equation are:

[cos x pm frac{1}{sqrt{2}} pm frac{sqrt{3}}{2}]

Which translates to:

[x pmfrac{pi}{4}, pmfrac{3pi}{4}, pmfrac{5pi}{4}, pmfrac{7pi}{4}, pmfrac{pi}{6}, pmfrac{5pi}{6}, pmfrac{7pi}{6}, pmfrac{11pi}{6} text{and} x pmfrac{pi}{2}, pmfrac{3pi}{2} text{for} x in [0, 2pi]]

Conclusion

Both methods used in this article lead to the same set of solutions, confirming the validity of our approach. Trigonometric equations can be approached in various ways, and understanding multiple techniques can greatly enhance your problem-solving skills in mathematics.

Moving forward, these identities and techniques are essential tools for solving more complex trigonometric equations in advanced mathematics. Practice and understanding the underlying principles will help you master these concepts.