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

Options

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

Answer

To solve the equation \(1 + \sin \theta + \sin^2 \theta + \sin^3 \theta + \ldots = 4 + 2\sqrt{3}\), we can follow these steps:

### Step 1: Identify the series
The left-hand side is an infinite geometric series where the first term \(a = 1\) and the common ratio \(r = \sin \theta\).

### Step 2: Use the formula for the sum of an infinite geometric series
The sum \(S\) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values we have:
\[ S = \frac{1}{1 - \sin \theta} \]

### Step 3: Set the equation
We know from the problem statement that this sum is 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 gives:
\[ (4 + 2\sqrt{3})\sin \theta = (4 + 2\sqrt{3}) - 1 \]
\[ (4 + 2\sqrt{3})\sin \theta = 3 + 2\sqrt{3} \]

### 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, we can multiply the numerator and the 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,
\[ \sin \theta = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

### Step 9: Find \(\theta\)
The values of \(\theta\) for which \(\sin \theta = \frac{\sqrt{3}}{2}\) in the interval \(0 < \theta < \pi\) are:
\[ \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{2\pi}{3} \]

### Final Answer
Thus, the values of \(\theta\) are:
\[ \theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{2\pi}{3} \]