If ` r sin theta = sqrt3, r + 4 sin theta = 2 (sqrt3 + 1), 0 le theta le 2pi`, then `theta`=

Options

A. `30^@`
B. `60^@`
C. `120^@`
D. `150^@`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Set up the equations
We are given two equations:
1. \( r \sin \theta = \sqrt{3} \)
2. \( r + 4 \sin \theta = 2(\sqrt{3} + 1) \)

### Step 2: Express \( r \) in terms of \( \sin \theta \)
From the first equation, we can express \( r \):
\[ r = \frac{\sqrt{3}}{\sin \theta} \]

### Step 3: Substitute \( r \) into the second equation
Now, substitute \( r \) into the second equation:
\[ \frac{\sqrt{3}}{\sin \theta} + 4 \sin \theta = 2(\sqrt{3} + 1) \]

### Step 4: Multiply through by \( \sin \theta \) to eliminate the fraction
Multiply the entire equation by \( \sin \theta \):
\[ \sqrt{3} + 4 \sin^2 \theta = 2(\sqrt{3} + 1) \sin \theta \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 4 \sin^2 \theta - 2\sqrt{3} \sin \theta + \sqrt{3} = 0 \]

### Step 6: Solve the quadratic equation
This is a quadratic equation in terms of \( \sin \theta \). We can use the quadratic formula:
\[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4, b = -2\sqrt{3}, c = \sqrt{3} \).

Calculating the discriminant:
\[ b^2 - 4ac = (-2\sqrt{3})^2 - 4 \cdot 4 \cdot \sqrt{3} = 12 - 16\sqrt{3} \]

### Step 7: Find the roots
Now substituting back into the quadratic formula:
\[ \sin \theta = \frac{2\sqrt{3} \pm \sqrt{12 - 16\sqrt{3}}}{8} \]

### Step 8: Simplify the roots
We can simplify this expression further, but we need to find specific values for \( \theta \).

### Step 9: Find specific angles
We know that:
1. \( \sin \theta = \frac{1}{2} \) gives us \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \)
2. \( \sin \theta = \frac{\sqrt{3}}{2} \) gives us \( \theta = \frac{\pi}{3}, \frac{2\pi}{3} \)

### Step 10: List all possible values of \( \theta \)
Thus, the possible values of \( \theta \) in the interval \( [0, 2\pi] \) are:
- \( \frac{\pi}{6} \) (30 degrees)
- \( \frac{5\pi}{6} \) (150 degrees)
- \( \frac{\pi}{3} \) (60 degrees)
- \( \frac{2\pi}{3} \) (120 degrees)

### Final Answer
The values of \( \theta \) are:
\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{\pi}{3}, \frac{2\pi}{3} \]