To solve the equation [tex]\(\cos(x) + \cos(2x) + \cos(3x) = 0\)[/tex], let's explore the process step-by-step.
1. Identify Key Trigonometric Identities:
Consider using trigonometric identities and known properties of cosine to simplify the equation.
2. Symmetry and Periodicity:
Recall the cosine function's periodicity [tex]\(\cos(x + 2\pi) = \cos(x)\)[/tex]. This periodic property means that solutions may repeat after every [tex]\(2\pi\)[/tex].
3. Evaluate Combination of Angles:
We look for specific angles that satisfy the given equation within one period, i.e., from [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex]. We assemble various standard values where [tex]\(\cos(\theta)\)[/tex] may yield simple values like [tex]\(0, \pm1, \pm\frac{1}{2}, \pm\frac{\sqrt{3}}{2}\)[/tex].
4. Analytical or Graphical Insight:
By exploring particular values of [tex]\(x\)[/tex] within the intervals and evaluating [tex]\(\cos(x)\)[/tex], [tex]\(\cos(2x)\)[/tex], and [tex]\(\cos(3x)\)[/tex], we ascertain if their sum is zero. Comprehensive coverage of such values for [tex]\(x\)[/tex] might be derived through factoring, setting up quadratic forms, or leveraging sophisticated computational tools.
Following these approaches, the specific solutions that satisfy [tex]\(\cos(x) + \cos(2x) + \cos(3x) = 0\)[/tex] are:
[tex]\[ x = -\frac{3\pi}{4}, \, -\frac{2\pi}{3}, \, -\frac{\pi}{4}, \, \frac{\pi}{4}, \, \frac{2\pi}{3}, \, \frac{3\pi}{4} \][/tex]
These roots are specifically derived at angles where the sum of the cosine terms evaluates to zero, considering the periodic and symmetric nature of the trigonometric functions involved.
