If `1 + sintheta + sin^2theta + sin^3theta +.. oo = 4 + 2sqrt3, 0 lt theta lt pi, theta != pi/2` then

Options

A. `theta = pi/3`
B. `theta = pi/6`
C. `theta = pi/3 or pi/6`
D. `theta = pi/3 or (2pi)/3`

Answer

To solve the problem, we need to find the values of \( \theta \) such that

\[ 1 + \sin \theta + \sin^2 \theta + \sin^3 \theta + \ldots = 4 + 2\sqrt{3} \]

### Step 1: Identify the series as a geometric series

The left-hand side of the equation is an infinite geometric series where:
- The first term \( a = 1 \)
- The common ratio \( r = \sin \theta \)

The sum \( S \) of an infinite geometric series can be calculated using the formula:

\[ S = \frac{a}{1 - r} \]

### Step 2: Apply the formula for the sum of the series

Substituting the values of \( a \) and \( r \):

\[ S = \frac{1}{1 - \sin \theta} \]

### Step 3: Set the sum equal to the given value

Now, we set the sum equal to \( 4 + 2\sqrt{3} \):

\[ \frac{1}{1 - \sin \theta} = 4 + 2\sqrt{3} \]

### Step 4: Cross-multiply to eliminate the fraction

Cross-multiplying gives:

\[ 1 = (4 + 2\sqrt{3})(1 - \sin \theta) \]

### Step 5: Expand the right-hand side

Expanding the right-hand side:

\[ 1 = (4 + 2\sqrt{3}) - (4 + 2\sqrt{3})\sin \theta \]

### Step 6: Rearrange the equation

Rearranging the equation gives:

\[ (4 + 2\sqrt{3})\sin \theta = (4 + 2\sqrt{3}) - 1 \]

This simplifies to:

\[ (4 + 2\sqrt{3})\sin \theta = 3 + 2\sqrt{3} \]

### Step 7: Solve for \( \sin \theta \)

Dividing both sides by \( 4 + 2\sqrt{3} \):

\[ \sin \theta = \frac{3 + 2\sqrt{3}}{4 + 2\sqrt{3}} \]

### Step 8: Rationalize the denominator

To simplify \( \sin \theta \), we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \sin \theta = \frac{(3 + 2\sqrt{3})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})} \]

Calculating the denominator:

\[ (4 + 2\sqrt{3})(4 - 2\sqrt{3}) = 16 - 12 = 4 \]

Calculating the numerator:

\[ (3 + 2\sqrt{3})(4 - 2\sqrt{3}) = 12 - 6\sqrt{3} + 8\sqrt{3} - 12 = 2\sqrt{3} \]

Thus, we have:

\[ \sin \theta = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

### Step 9: Find the values of \( \theta \)

The values of \( \theta \) for which \( \sin \theta = \frac{\sqrt{3}}{2} \) are:

\[ \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{2\pi}{3} \]

### Conclusion

Thus, the required values of \( \theta \) are:

\[ \theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{2\pi}{3} \]