If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:

Question

Let \( \cos(\alpha + \beta) = -\dfrac{1}{10} \) and \( \sin(\alpha - \beta) = \dfrac{3}{8} \), where \( 0<\alpha<\dfrac{\pi}{3} \) and \( 0<\beta<\dfrac{\pi}{4} \).
If \[ \tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}, \] then the value of \( r + s \) is ______________.

Answer 1

The provided equation is: \[\sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0\]Substituting \( x = \csc \theta \), the equation transforms into a quadratic form: \[\sqrt{3} x^2 - 2 (\sqrt{3} - 1) x - 4 = 0\]Applying the quadratic formula to solve for \( x \): \[x = \frac{-(-2 (\sqrt{3} - 1)) \pm \sqrt{(-2 (\sqrt{3} - 1))^2 - 4 \cdot \sqrt{3} \cdot (-4)}}}{2 \cdot \sqrt{3}}\]After simplifying the discriminant and calculating the values of \( x \), we find two distinct real solutions for \( x \). This yields two possible values for \( \csc \theta \), which in turn result in 3 solutions for \( \theta \) within the specified interval.Consequently, the total number of solutions is 3.

If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:

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The Correct Option is C

Solution and Explanation

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