Question textSolve the equation [math: −2sin⁡(x)−3=0]{-2\,\sin \left( x \right)-\sqrt{3}=0} for [math: x∈[0,2π]]\displaystyle x\in \Big[0, 2\pi \Big], giving your answer in radians. Note. Type pi to enter [math: π]\pi and write your answers [math: a] and [math: b] in the form [math: {a,b}]\{a,b\}. [input] Check Question 3

Question

BlackTom AI · Accepted Answer

The prompt presents a math problem asking to solve an equation for x in [0, 2π] and to enter the answer in radians, formatted as {a,b} using pi notation.

First, I restate the core equation(s) as they appear: one version reads -2 sin(x) - 3 = 0, and another reads -2 sin(x) - sqrt(3) = 0. These are not the same equation, so it's important to identify which is intended for solving.

If we take the first version, -2 sin(x) - 3 = 0, we would isolate sin(x): -2 sin(x) = 3, so sin(x) = -3/2. This value is impossible because the sine function ranges between -1 and 1. Therefore, there is no solution in [0, 2π] for this version.

If we take the second version, -2 sin(x) - sqrt(3) = 0, we get -2 sin(x) = sqrt(3), so sin(x) = -sqrt(3)/2. This is a valid sine value. Now we look for angles in [0, 2π] with sine equal to -√3/2. The standard reference angle for √3/2 is π/3 (or 60 degrees). Sine is negative in the third and fourth quadrants, so the corresponding angles are: x = π + π/3 = 4π/3, and x = 2π - π/3 = 5π/3.

Thus, the solutions in [0, 2π] are x ∈ {4π/3, 5π/3}.

Now addressing the “answer_options” field: this particular item shows an empty list of options, and the “answer” field provides the solution {4π/3, 5π/3}. Since there are no explicit multiple-choice options to evaluate, there is nothing to analyze for each option in the sense of choosing among disruptors; however, we can still justify the provided answer by the reasoning above.

If you intended to present multiple choices for a student to select from, you would include options such as:
- A) {π/3, 5π/3}
- B) {4π/3, 5π/3}
- C) {π/6, 7π/6}
- D) No solution
Then Option B would be correct, consistent with sin(x) = -√3/2 in the interval [0, 2π].

In summary: the solution set {4π/3, 5π/3} corresponds to solving sin(x) = -√3/2 within the interval [0, 2π], while the equation -2 sin(x) - 3 = 0 has no solution because it requests sin(x) = -3/2, which is outside the range [-1, 1].

MCD2130 - T2 - 2025 MCD2130 Sample Test 3

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