If-1-sin-x-sin-2-x-sin-3-x-4-2-3-0-lt-x-lt-pi-Find-the-value-of-x-

Question Number 177933 by Spillover last updated on 11/Oct/22

If 1 + \sin x + \sin^{2} x + \sin^{3} x + \dots \infty

= 4 + 2 \sqrt{3}, 0 < x < π Find the value of x

Answered by Ar Brandon last updated on 11/Oct/22

1 + \sin x + \sin^{2} x + \sin^{3} x + \cdot \cdot \cdot = \frac{1}{1 - \sin x} = 4 + 2 \sqrt{3}

\Rightarrow \frac{1}{1 - \sin x} = 2 (2 + \sqrt{3}) = \frac{2}{2 - \sqrt{3}} = \frac{1}{1 - \frac{\sqrt{3}}{2}}

\Rightarrow \sin x = \frac{\sqrt{3}}{2} \Rightarrow x = \frac{π}{3}, \frac{2 π}{3}

Commented by Spillover last updated on 11/Oct/22

t h a n k s f o r s o l v i n g

Answered by Rasheed.Sindhi last updated on 11/Oct/22

1 + \sin x + \sin^{2} x + \sin^{3} x + \dots \infty = 4 + 2 \sqrt{3}

\sin x + \sin^{2} x + \sin^{3} x + \dots \infty = 3 + 2 \sqrt{3}

\sin x (1 + \sin x + \sin^{2} x + \sin^{3} x + \dots \infty) = 3 + 2 \sqrt{3}

\sin x (4 + 2 \sqrt{3}) = 3 + 2 \sqrt{3}

\sin x = \frac{3 + 2 \sqrt{3}}{4 + 2 \sqrt{3}} \cdot \frac{4 - 2 \sqrt{3}}{4 - 2 \sqrt{3}} = \frac{12 - 4 (3) + 8 \sqrt{3} - 6 \sqrt{3}}{16 - 4 (3)}

\sin x = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} \Rightarrow x = 60 °, 120 °

Commented by Spillover last updated on 11/Oct/22

nice solution