The principal solution `sqrt(3)cosecx+2=0` are

Options

A. `(pi)/(3), (3pi)/(3)`
B. `(2pi)/(3), (5pi)/(3)`
C. `(4pi)/(3), (5pi)/(3)`
D. `(pi)/(3), (4pi)/(3)`

Answer

To solve the equation \( \sqrt{3} \cos x + 2 = 0 \) for the principal solutions, we can follow these steps:

### Step 1: Isolate \( \cos x \)
We start with the equation:
\[ \sqrt{3} \cos x + 2 = 0 \]
Subtract 2 from both sides:
\[ \sqrt{3} \cos x = -2 \]
Now, divide both sides by \( \sqrt{3} \):
\[ \cos x = -\frac{2}{\sqrt{3}} \]

### Step 2: Simplify the expression
To simplify \( -\frac{2}{\sqrt{3}} \), we can rationalize the denominator:
\[ \cos x = -\frac{2 \sqrt{3}}{3} \]

### Step 3: Determine the reference angle
The reference angle \( \theta \) for \( \cos x = \frac{2}{\sqrt{3}} \) is:
\[ \theta = \frac{\pi}{3} \]
This is because \( \cos \frac{\pi}{3} = \frac{1}{2} \) and we need to find the angle where the cosine is negative.

### Step 4: Find the angles in the correct quadrants
Since cosine is negative in the second and third quadrants, we can find the angles:
1. In the second quadrant:
\[ x = \pi - \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]
2. In the third quadrant:
\[ x = \pi + \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \]

### Step 5: Write the principal solutions
Thus, the principal solutions for the equation \( \sqrt{3} \cos x + 2 = 0 \) are:
\[ x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3} \]

### Final Answer
The principal solutions are \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \).

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