Algebraic Relationship Analysis

The problem asks us to find the value of the expression $x^3 + \frac{1}{x^3}$, given a specific relationship between $x$ and $\theta$: $x + \frac{1}{x} = 2\cos\theta$ We need to use this given equation to determine the value of $x^3 + \frac{1}{x^3}$.

Trigonometric Representation using Euler's Formula

A common technique to solve problems relating algebraic expressions like $x + \frac{1}{x}$ to trigonometric functions is to use Euler's formula, which connects complex exponentials to trigonometric functions. Euler's formula states: $e^{i\phi} = \cos\phi + i\sin\phi$ Let's consider if we can represent $x$ in a form that satisfies the given condition $x + \frac{1}{x} = 2\cos\theta$. Suppose we choose $x$ to be: $x = e^{i\theta} = \cos\theta + i\sin\theta$ If this is the case, then $\frac{1}{x}$ would be: $\frac{1}{x} = x^{-1} = e^{-i\theta} = \cos(-\theta) + i\sin(-\theta) = \cos\theta - i\sin\theta$ Now, let's check if this choice of $x$ satisfies the given condition:

• Calculate $x + \frac{1}{x}$: $x + \frac{1}{x} = (\cos\theta + i\sin\theta) + (\cos\theta - i\sin\theta)$ $x + \frac{1}{x} = 2\cos\theta$

This matches the given equation, so our representation $x = e^{i\theta}$ is valid for this problem.

Calculating $x^3 + \frac{1}{x^3}$

Now that we have established $x = e^{i\theta}$, we can easily find $x^3$ and $\frac{1}{x^3}$.

• Calculate $x^3$: Using the property $(e^{a})^b = e^{ab}$, we get: $x^3 = (e^{i\theta})^3 = e^{i(3\theta)}$ Applying Euler's formula again: $x^3 = \cos(3\theta) + i\sin(3\theta)$
• Calculate $\frac{1}{x^3}$: Similarly, $\frac{1}{x^3} = x^{-3} = (e^{i\theta})^{-3} = e^{-i(3\theta)}$ Applying Euler's formula: $\frac{1}{x^3} = \cos(-3\theta) + i\sin(-3\theta) = \cos(3\theta) - i\sin(3\theta)$
• Calculate the sum $x^3 + \frac{1}{x^3}$: $x^3 + \frac{1}{x^3} = (\cos(3\theta) + i\sin(3\theta)) + (\cos(3\theta) - i\sin(3\theta))$ $x^3 + \frac{1}{x^3} = 2\cos(3\theta)$

Conclusion

By representing $x$ using Euler's formula as $x = e^{i\theta}$, which satisfies the given condition $x + \frac{1}{x} = 2\cos\theta$, we found that $x^3 + \frac{1}{x^3}$ simplifies to $2\cos(3\theta)$. This matches option 3.