Verified Answer :

  • Isolate the trigonometric term: \(\sin(2x) = \frac{3}{5}\).
  • Define \(\theta = 2x\), where \(\theta \in [0, 4\pi)\) for \(x \in [0, 2\pi)\).
  • Identify that \(\sin(\theta) = 0.6\) has two solutions in every period of \(2\pi\), yielding 4 solutions in total over \([0, 4\pi)\).
  • Convert back to \(x\) to confirm there are \(\boxed{4}\) distinct solutions.

Explanation

  1. Isolating the Sine Function

We are given the trigonometric equation \(5 \sin(2x) + 1 = 4\) and we need to determine the number of distinct solutions in the interval \(x \in [0, 2\pi)\). We start by isolating the trigonometric term \(\sin(2x)\) in the equation:
\[5 \sin(2x) + 1 = 4\]
Subtracting 1 from both sides gives:
\[5 \sin(2x) = 3\]
Dividing by 5 results in:
\[\sin(2x) = \frac{3}{5}\]
This equation holds when \(2x\) corresponds to an angle whose sine is \(0.6\).

  1. Solving for \(\theta\) in the Extended Interval

Let \(\theta = 2x\). Since \(x \in [0, 2\pi)\), the variable \(\theta\) ranges over the interval \([0, 4\pi)\). We are solving \(\sin(\theta) = \frac{3}{5}\) for \(\theta \in [0, 4\pi)\).

The sine function \(\sin(\theta) = 0.6\) has two solutions within the first full period \([0, 2\pi)\):

  1. \(\theta_1 = \arcsin(0.6) \approx 0.6435\) radians.
  2. \(\theta_2 = \pi - \arcsin(0.6) \approx 2.4981\) radians.

Since the period of the sine function is \(2\pi\), we find additional solutions by adding \(2\pi\) to the solutions in the first period:

  1. \(\theta_3 = \theta_1 + 2\pi = 2\pi + \arcsin(0.6) \approx 6.9267\) radians.
  2. \(\theta_4 = \theta_2 + 2\pi = 3\pi - \arcsin(0.6) \approx 8.7813\) radians.

These four values for \(\theta\) are all distinct and lie within the interval \([0, 4\pi)\).

  1. Determining the Solutions for \(x\) and Final Counting

Now we map these solutions back to \(x\) using \(x = \frac{\theta}{2}\):

  1. \(x_1 = \frac{\arcsin(0.6)}{2} \approx 0.3217\)
  2. \(x_2 = \frac{\pi - \arcsin(0.6)}{2} \approx 1.2490\)
  3. \(x_3 = \frac{2\pi + \arcsin(0.6)}{2} = \pi + \frac{\arcsin(0.6)}{2} \approx 3.4633\)
  4. \(x_4 = \frac{3\pi - \arcsin(0.6)}{2} = 1.5\pi - \frac{\arcsin(0.6)}{2} \approx 4.3907\)

All four values fall within the interval \(x \in [0, 2\pi)\), where \(2\pi \approx 6.283\). Thus, there are 4 distinct solutions to the equation. The correct option is B.

Examples

In engineering, this problem is akin to determining the moments when a mechanical wave or an oscillating signal reaches a certain intensity threshold. If a sensor is calibrated to detect when a signal \(5 \sin(2x) + 1\) hits a value of \(4\), the solution shows there are exactly four specific time instances within one cycle (represented by \(2\pi\)) where this occurs. Such calculations are essential for timing signal sampling, optimizing mechanical vibrations, or ensuring electronic components operate within safe thresholds.

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