Question

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

• A. 6
• B. 8
• C. 10
• D. 7

Hint:

In trigonometric equations, look for multiple possible values of \( \sin\theta \) and count the number of solutions within the given range.

Answer 1

The Correct Option is A

Determine the count of solutions for the trigonometric equation \( \sqrt{3} \csc^2\theta - 2(\sqrt{3} - 1) \csc\theta - 4 = 0 \) within the interval \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \).

Underlying Principle:

The provided equation is a quadratic in \( \csc\theta \). It can be solved by applying the quadratic formula to find values of \( \csc\theta \). The formula for the roots of \( ax^2 + bx + c = 0 \) is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

After obtaining the values for \( \csc\theta \), we find the corresponding \( \sin\theta \) values. Subsequently, all angles \( \theta \) that satisfy these values and fall within the specified range \( \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \) are identified.

Solution Breakdown:

Step 1: Solve the quadratic equation for \( \csc\theta \).

Let \( x = \csc\theta \). The equation transforms into \( \sqrt{3}x^2 - 2(\sqrt{3} - 1)x - 4 = 0 \). The coefficients are \( a = \sqrt{3} \), \( b = -2(\sqrt{3} - 1) \), and \( c = -4 \). The quadratic formula is used to find \(x\).

Step 2: Compute the discriminant (\( \Delta = b^2 - 4ac \)).

\[ \Delta = \left(-2(\sqrt{3} - 1)\right)^2 - 4(\sqrt{3})(-4) \] \[ = 4(\sqrt{3}^2 - 2\sqrt{3} + 1) + 16\sqrt{3} \] \[ = 4(3 - 2\sqrt{3} + 1) + 16\sqrt{3} \] \[ = 4(4 - 2\sqrt{3}) + 16\sqrt{3} \] \[ = 16 - 8\sqrt{3} + 16\sqrt{3} = 16 + 8\sqrt{3} \]

The square root of the discriminant is simplified:

\[ 16 + 8\sqrt{3} = 4(4 + 2\sqrt{3}) = 4(3 + 1 + 2\sqrt{3}) = 4(\sqrt{3} + 1)^2 \]

Therefore, \( \sqrt{\Delta} = \sqrt{4(\sqrt{3} + 1)^2} = 2(\sqrt{3} + 1) \).

Step 3: Determine the two possible values for \( \csc\theta \).

\[ \csc\theta = \frac{-[-2(\sqrt{3} - 1)] \pm 2(\sqrt{3} + 1)}{2\sqrt{3}} \] \[ \csc\theta = \frac{2(\sqrt{3} - 1) \pm 2(\sqrt{3} + 1)}{2\sqrt{3}} = \frac{(\sqrt{3} - 1) \pm (\sqrt{3} + 1)}{\sqrt{3}} \]

This yields two distinct cases:

Case 1 (using the '+' operator):

\[ \csc\theta = \frac{(\sqrt{3} - 1) + (\sqrt{3} + 1)}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \]

Case 2 (using the '-' operator):

\[ \csc\theta = \frac{(\sqrt{3} - 1) - (\sqrt{3} + 1)}{\sqrt{3}} = \frac{-2}{\sqrt{3}} \]

Step 4: Identify solutions for each case within the interval \( \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \).

For Case 1: \( \csc\theta = 2 \implies \sin\theta = \frac{1}{2} \). Principal solutions for \( \theta \) are \( \frac{\pi}{6} \) and \( \pi - \frac{\pi}{6} = \frac{5\pi}{6} \). Within the interval \( \left[ -210^\circ, 240^\circ \right] \), the solutions are:

• \( \theta = \frac{\pi}{6} \)
• \( \theta = \frac{5\pi}{6} \)
• Considering negative angles: \( \frac{5\pi}{6} - 2\pi = -\frac{7\pi}{6} \), which is at the lower boundary.

Thus, Case 1 provides three solutions: \( -\frac{7\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6} \).

For Case 2: \( \csc\theta = -\frac{2}{\sqrt{3}} \implies \sin\theta = -\frac{\sqrt{3}}{2} \). Principal solutions in \( [-\pi, \pi] \) are \( -\frac{\pi}{3} \) and \( -\frac{2\pi}{3} \). Within the interval \( \left[ -210^\circ, 240^\circ \right] \), the solutions are:

• \( \theta = -\frac{\pi}{3} \)
• \( \theta = -\frac{2\pi}{3} \)
• Checking positive angles: \( \pi + \frac{\pi}{3} = \frac{4\pi}{3} \), which is at the upper boundary. \( 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \) is outside the interval.

Thus, Case 2 provides three solutions: \( -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{4\pi}{3} \).

 

Step 5: Aggregate the total number of distinct solutions.

The solutions from Case 1 are \( \left\{ -\frac{7\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6} \right\} \). The solutions from Case 2 are \( \left\{ -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{4\pi}{3} \right\} \). All these six solutions are unique and within the specified interval. The total count of solutions is 3 + 3 = 6.

The equation has a total of 6 solutions.