Question

Number of solutions of the equation \[ \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0 \] in \( \theta \in \left[ -3\pi, 2\pi \right] \):

Hint:

For trigonometric equations involving double angles, use trigonometric identities to simplify the equation and then solve for the variable using the quadratic formula if applicable.

Answer 1

Correct Answer: 5

Step 1: Express the equation in a simpler form.
We are given the equation: \[ \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0 \] Using the double angle identity \( \cos 2\theta = 2 \cos^2 \theta - 1 \), substitute into the equation: \[ \sqrt{3}(2 \cos^2 \theta - 1) + 8 \cos \theta + 3\sqrt{3} = 0 \] Simplifying the equation, we get: \[ 2\sqrt{3} \cos^2 \theta - \sqrt{3} + 8 \cos \theta + 3\sqrt{3} = 0 \] \[ 2\sqrt{3} \cos^2 \theta + 8 \cos \theta + 2\sqrt{3} = 0 \]
Step 2: Solve the quadratic equation.
This is a quadratic equation in terms of \( \cos \theta \). Let \( x = \cos \theta \), the equation becomes: \[ 2\sqrt{3} x^2 + 8x + 2\sqrt{3} = 0 \] Solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2\sqrt{3}, b = 8, \text{ and } c = 2\sqrt{3} \). Substitute these values into the quadratic formula and calculate the values of \( x \).
Step 3: Find the number of solutions.
After solving for \( x \), calculate the corresponding values of \( \theta \) within the given interval \( \left[ -3\pi, 2\pi \right] \).
Step 4: Conclusion.
The total number of solutions for \( \theta \) is 5. Final Answer: \[ \boxed{5} \]