How to Solve Equations Involving Trigonometric and Exponential Functions

In this article, we will explore a specific equation that combines trigonometric and exponential functions. We will break down the process of solving such an equation step-by-step, highlighting the key characteristics of these functions and how they interact with each other.

Equation to Solve

The given equation is:
cos^2(x23) 3x3-x2

Analysis and Solution Steps

Step 1: Simplifying and Understanding the Equation

Firstly, let's simplify the equation and understand its components:

cos^{2} left(frac{x^2}{3}right) frac{3^x 3^{-x}}{2}

We know that:

cos^{2} theta leq 1 a 3^x
Thus,
frac{a frac{1}{a}}{2} leq 1

From the above, we can infer that:

a leq 1 implies x leq 0

The only value that satisfies both conditions is x0.

Step 2: Proving the Right Hand Side is 2

Let's prove the right hand side (RHS) of the equation is equal to 2:

Consider f(t) frac{t}{t-1} where t geq 0.

At:

f(0) rightarrow infty f(infty) rightarrow 1

Derivative of f(t):

f'(t) frac{1-frac{1}{t^2}}{1}0 implies t1

The minimum value of f(t) is 2 when t1.

Step 3: Analyzing the Left Hand Side (LHS)

On the other hand, the left hand side (LHS) must be within the range 0 to 2. Particularly, it achieves 2 at x0:

2cos^{2}left(frac{0^2}{3}right) 2cos^{2}0 2

Thus, the only solution is:

Conclusion

In conclusion, the solution to the equation
2cos^{2}left(frac{x^{2}x}{3}right) 3^{x} 3^{-x}
is

Understanding the behavior and properties of trigonometric and exponential functions is crucial to solving such equations effectively.