Verified Answer :

  • Isolate the trigonometric term: \(\cos(3x) = -\frac{1}{6}\).
  • Define a new variable \(u = 3x\) such that \(0 \leq u < 6\pi\).
  • Recognize that \(\cos(u) = -\frac{1}{6}\) has two solutions in each \(2\pi\) interval.
  • Calculate the total: \(3 \text{ periods} \times 2 \text{ solutions/period} = 6\) solutions.
  • The final answer is \(\boxed{A}\).

Explanation

  1. Isolating the trigonometric term

The given equation is \(6 \cos(3x) + 1 = 0\), and we are interested in finding the number of distinct solutions for \(x\) in the interval \([0, 2\pi)\). First, we isolate the trigonometric term by subtracting 1 from both sides and then dividing by 6, which yields \(\cos(3x) = -\frac{1}{6}\).

  1. Changing variables to define the new interval

Let \(u = 3x\). Given that \(x\) lies in the interval \([0, 2\pi)\), we can determine the range for \(u\) by multiplying the entire interval by 3:
\[0 \le x < 2\pi \implies 0 \le 3x < 6\pi \implies 0 \le u < 6\pi.\]
Our problem is now equivalent to finding the number of solutions for the equation \(\cos(u) = -\frac{1}{6}\) within the interval \([0, 6\pi)\).

  1. Counting the solutions in the interval

The cosine function, \(\cos(u)\), has a period of \(2\pi\). This means it completes one full cycle every \(2\pi\) interval. The interval \([0, 6\pi)\) consists of exactly three periods: \([0, 2\pi)\), \([2\pi, 4\pi)\), and \([4\pi, 6\pi)\). In any interval of the form \([2k\pi, 2(k+1)\pi)\), the equation \(\cos(u) = C\) (where \(-1 < C < 1\)) has exactly two solutions. Since \(-\frac{1}{6}\) satisfies \(-1 < -\frac{1}{6} < 1\), there are two solutions for \(u\) in each of the three full periods.

  1. Final calculation and conclusion

By multiplying the number of full periods by the number of solutions per period, we get the total number of solutions for \(u\):
\[3 \text{ periods} \times 2 \text{ solutions/period} = 6 \text{ solutions}.\]
Since each solution for \(u\) uniquely corresponds to a solution for \(x\) via the linear mapping \(u = 3x\), there are 6 distinct solutions for \(x\) in the original interval \([0, 2\pi)\). Thus, the correct option is A.

Examples

In signal processing, calculating how often a waveform hits a specific threshold is essential for analyzing frequency components. For instance, if you have a periodic vibration modeled by \(6 \cos(3x) + 1 = 0\), knowing the number of times it crosses a threshold (like displacement zero relative to a baseline) helps engineers understand the system's behavior over a set time window. This is a common application of solving trigonometric equations to determine state changes in physical or electrical systems.

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