Question

If 1 + sin θ + sin² θ + ... upto ∞ = 2√3 + 4, then θ = ______
1. 3π/4
2. π/3
3. π/4
4. π/6

Answer 1

Correct Answer - Option 2 : π/3

Concept:

The sum of an infinite G.P. is 

S = \(\rm a\over 1-r\)

Where a is the first term of the series and r is the common ratio

Calculation:

Given 1 + sin θ + sin2 θ + ... upto ∞ = 2√3 + 4

First term of the G.P is a = 1 and r = sin θ 

∴ \(\rm 1\over 1 - \sin θ\) = 2√3 + 4

1 - sin θ = \(1\over2\sqrt3 + 4\)

1 - sin θ = \({1\over2\sqrt3 + 4} \times {2\sqrt3 - 4\over 2\sqrt3 - 4}\)

1 - sin θ = \( {4-2\sqrt3 \over 16-12}\)

1 - sin θ = \( 1-{\sqrt3 \over 2}\)

sin θ = \( {\sqrt3 \over 2}\) 

⇒ θ = \(\boldsymbol{\pi\over3}\)