Question

The number of solutions of the equation \( (4 - \sqrt{3}) \sin x - 2\sqrt{3} \cos^2 x = \frac{-4}{1 + \sqrt{3}} \), \( x \in \left[-2\pi, \frac{5\pi}{2}\right] \) is

• A. 4
• B. 3
• C. 6
• D. 5

Hint:

Convert the given trigonometric equation into a quadratic equation in \( \sin x \) or \( \cos x \). Solve the quadratic equation to find the possible values of the trigonometric function. Then, find the number of solutions for these values within the specified interval. Pay careful attention to the boundaries of the interval.

Answer 1

The Correct Option is D

To determine the number of solutions for the equation \( (4 - \sqrt{3}) \sin x - 2\sqrt{3} \cos^2 x = \frac{-4}{1 + \sqrt{3}} \) in the interval \( x \in \left[-2\pi, \frac{5\pi}{2}\right] \), we proceed as follows:

1. Simplify the right-hand side of the equation:

\[\frac{-4}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{-4(1 - \sqrt{3})}{1 - 3} = \frac{-4 + 4\sqrt{3}}{-2} = 2 - 2\sqrt{3}\]

1. The equation transforms to:

\[(4 - \sqrt{3}) \sin x - 2\sqrt{3} \cos^2 x = 2 - 2\sqrt{3}\]

1. Substitute \(\cos^2 x = 1 - \sin^2 x\):

\[(4 - \sqrt{3}) \sin x - 2\sqrt{3} (1 - \sin^2 x) = 2 - 2\sqrt{3}\]

1. Expand and simplify:

\[(4 - \sqrt{3}) \sin x - 2\sqrt{3} + 2\sqrt{3} \sin^2 x = 2 - 2\sqrt{3}\]

1. Rearrange terms to group \(\sin x\) and constants:

\[2\sqrt{3} \sin^2 x + (4 - \sqrt{3}) \sin x = 0\]

1. Factor out \(\sin x\):

\[\sin x (2\sqrt{3} \sin x + 4 - \sqrt{3}) = 0\]

1. This yields two separate equations:
   • Equation 1: \(\sin x = 0\)
   • Equation 2: \(2\sqrt{3} \sin x + 4 - \sqrt{3} = 0\)
2. For \(\sin x = 0\), the solutions in the interval \([−2\pi, \frac{5\pi}{2}]\) are:

\[x = -2\pi, 0, \pi, 2\pi\]

1. Solve Equation 2 for \(\sin x\):

\[2\sqrt{3} \sin x = \sqrt{3} - 4\]\[\sin x = \frac{\sqrt{3} - 4}{2\sqrt{3}}\]

1. Approximate the value of \(\sin x\):

\[\sin x \approx -0.633\]

1. Determine the solutions for \(\sin x \approx -0.633\) within the given domain. This involves finding angles where the sine function equals this negative value, considering the periodicity and the specified interval. The inverse sine function will yield a principal value, and other solutions can be found by adding or subtracting multiples of \(2\pi\) and considering the symmetry of the sine wave.
2. Consolidate all solutions:
   • Four solutions from \(\sin x = 0\): \(x = -2\pi, 0, \pi, 2\pi\).
   • An additional solution arising from \(\sin x = \frac{\sqrt{3} - 4}{2\sqrt{3}}\) within the interval \([-2\pi, \frac{5\pi}{2}]\).
3. The total number of solutions is \(\textbf{5}\).