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

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

Concept Used:

The given equation is a quadratic equation in terms of \( \csc\theta \). We can solve for \( \csc\theta \) by treating it as a variable in a standard quadratic equation \( ax^2 + bx + c = 0 \). The solutions for \(x\) are given by the quadratic formula:

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

Once we find the values of \( \csc\theta \), we determine the corresponding values of \( \sin\theta \). Finally, we find all angles \( \theta \) that satisfy these values and lie within the given interval \( \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \).

Step-by-Step Solution:

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

Let \( x = \csc\theta \). The equation becomes \( \sqrt{3}x^2 - 2(\sqrt{3} - 1)x - 4 = 0 \). Here, \( a = \sqrt{3} \), \( b = -2(\sqrt{3} - 1) \), and \( c = -4 \). We use the quadratic formula to find the values of \(x\).

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

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

To simplify the square root of the discriminant, we write it as a perfect square:

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

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

Step 3: Find 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 leads to two cases:

Case 1 (using the '+' sign):

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

Case 2 (using the '-' sign):

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

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

From Case 1: \( \csc\theta = 2 \implies \sin\theta = \frac{1}{2} \). The principal values for \( \theta \) are \( \frac{\pi}{6} \) and \( \pi - \frac{\pi}{6} = \frac{5\pi}{6} \). We check for solutions in the interval \( \left[ -210^\circ, 240^\circ \right] \). The solutions are:

• \( \theta = \frac{\pi}{6} \) (or \(30^\circ\))
• \( \theta = \frac{5\pi}{6} \) (or \(150^\circ\))
• We also check for negative angles: \( \frac{\pi}{6} - 2\pi = -\frac{11\pi}{6} \) (outside), and \( \frac{5\pi}{6} - 2\pi = -\frac{7\pi}{6} \) (or \(-210^\circ\)). This is at the boundary of the interval.

So, from this case, we have three solutions: \( -\frac{7\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6} \).

 

From Case 2: \( \csc\theta = -\frac{2}{\sqrt{3}} \implies \sin\theta = -\frac{\sqrt{3}}{2} \). The principal values for \( \theta \) (in the range \( [-\pi, \pi] \)) are \( -\frac{\pi}{3} \) and \( -\frac{2\pi}{3} \). We check for solutions in the interval \( \left[ -210^\circ, 240^\circ \right] \). The solutions are:

• \( \theta = -\frac{\pi}{3} \) (or \(-60^\circ\))
• \( \theta = -\frac{2\pi}{3} \) (or \(-120^\circ\))
• We check for positive angles: The solutions in \( [0, 2\pi] \) are \( \pi + \frac{\pi}{3} = \frac{4\pi}{3} \) (or \(240^\circ\)) and \( 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \) (outside). The value \( \frac{4\pi}{3} \) is at the boundary of the interval.

So, from this case, we have three solutions: \( -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{4\pi}{3} \).

 

Step 5: Count 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 solutions are distinct and lie within the given interval. Total number of solutions = 3 + 3 = 6.

Therefore, the total number of solutions is 6.