Question

If \( 1 + \sin\theta + \sin^2\theta + \dots \text{ upto } \infty = 2\sqrt{3} + 4 \), then \( \theta = \)

• A. \( \frac{3\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

To save time during the exam, you can substitute the options directly:
- For \( \theta = \frac{\pi}{3} \), \( \sin\theta = \frac{\sqrt{3}}{2} \).
- Sum \( S_\infty = \frac{1}{1 - \sqrt{3}/2} = \frac{2}{2 - \sqrt{3}} = 2(2 + \sqrt{3}) = 4 + 2\sqrt{3} \), which matches perfectly!

Answer 1

The Correct Option is B

Step 1: Recognise the infinite geometric series.
The left side $1+\sin\theta+\sin^2\theta+\cdots$ is a GP with first term $a=1$ and common ratio $r=\sin\theta$, valid since $|\sin\theta|<1$.
Step 2: Apply the sum formula.
\[ S_\infty = \frac{1}{1-\sin\theta} = 2\sqrt{3}+4 \]
Step 3: Simplify the right-hand value numerically.
Since $\sqrt{3}\approx 1.732$, we get $2\sqrt{3}+4 \approx 3.464+4 = 7.464$.
Step 4: Solve for $\sin\theta$.
Taking reciprocals, $1-\sin\theta = \dfrac{1}{7.464} \approx 0.134$, so $\sin\theta \approx 0.866$.
Step 5: Recognise the exact value.
$0.866$ is precisely $\dfrac{\sqrt{3}}{2}$, confirming $\sin\theta = \dfrac{\sqrt{3}}{2}$.
Step 6: Read off the angle.
Among the options, $\sin\theta=\dfrac{\sqrt 3}{2}$ corresponds to $\theta=\dfrac{\pi}{3}$. \[ \boxed{\theta = \dfrac{\pi}{3}} \]