Solving Complex Equations: A Comprehensive Approach Using Logarithms and Trigonometry

In this article, we explore a detailed and analytical approach to solving complex algebraic equations. We will use logarithmic and trigonometric functions to address the problem presented by the user. This comprehensive method will be broken down into simple steps to ensure clarity and ease of understanding.

Introduction

The problem to be addressed involves an equation that requires a deep understanding of multiple mathematical concepts. We start by simplifying the original equation and then proceed to solve it using logarithms and trigonometric identities. Our goal is to provide a clear and detailed solution to this challenging problem, which can be useful for students and professionals alike.

The Original Equation

The equation provided by the user is:

[frac{2^{sin x}}{3^{cos x}} frac{3}{4}]

We need to solve this equation for (x), which will require a multi-step approach involving logarithms and trigonometric identities.

Step 1: Simplifying the Equation

First, we simplify the given equation by isolating the terms in the numerator and the denominator:

[4 times 2^{sin x} 3 times 3^{cos x}]

We can further simplify this to:

[2^{2 sin x} 3^{1 - cos x}]

Step 2: Taking the Logarithm of Both Sides

To proceed, we take the logarithm of both sides of the equation:

[2 sin x cdot log 2 (1 - cos x) cdot log 3]

We then define (A log 2) and (B log 3).[A sin x - B cos x B - 2A]

Next, we rewrite the equation in a form that can be easier to solve:

[sqrt{A^2 B^2} sin x - alpha B - 2A]

where (sin alpha frac{B}{sqrt{A^2 B^2}}).

From here, we can use the general trigonometric solution for the sine function:

[x npi - (-1)^n beta - alpha]

where (sin beta frac{B - 2A}{sqrt{A^2 B^2}}) and (n cdots -2, -1, 0, 1, 2, cdots).

Step 3: Verification

It is important to check that the values of (sin alpha) and (sin beta) are legitimate by ensuring that:

(-1 leq frac{B}{sqrt{A^2 B^2}} leq 1) and (-1 leq frac{B - 2A}{sqrt{A^2 B^2}} leq 1).

Alternative Trigonometric Approach

Another approach involves rewriting the sine function and using the cosine function:

We let (y cos x), so the equation becomes:

[frac{2^{sqrt{1 - y^2}}}{3^y} frac{3}{4}]

Note that (3 2^{log 3}), so:

[2^{sqrt{1 - y^2} - log 3 y} frac{3}{4}]

Take logs of both sides:

[sqrt{1 - y^2} - log 3 y log 3 - 2]

Squaring both sides:

[1 - y^2 log^2 3 - 2y y^2 - 2]

Let (a log^2 3), then:

[1 - y^2 a(1 - 2y) 1 - a]

which simplifies to a quadratic equation:

[y frac{-2a pm sqrt{4a^2 12a - 3}}{2(1 - a)}]

Given that (a approx 2.51), solve for (y) and select the valid root:

[y frac{-5.02 pm sqrt{25.24 123.5}}{7.02} frac{-5.02 pm 8.2}{7.02}]

The valid solution is:

(y 0.455)

Conclusion

This comprehensive approach to solving the complex equation showcases the power of combining logarithms and trigonometry. By breaking down the problem into manageable steps, we have found a solution that is both accurate and insightful.

Keywords

logarithmic equations, trigonometric equations, goal seek