If-1-sinx-sin-2-x-sin-3-x-4-2-3-x-

Question Number 175904 by BaliramKumar last updated on 09/Sep/22

If 1 + s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty = 4 + 2 \sqrt{3},

x = ?

Commented by infinityaction last updated on 09/Sep/22

g . p . s u m

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

1 - \sin x = \frac{1}{4 + 2 \sqrt{3}} = \frac{4 - 2 \sqrt{3}}{16 - 12}

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

\sin x = \frac{\sqrt{3}}{2}

x = 60 °

Answered by Rasheed.Sindhi last updated on 09/Sep/22

AnOther way \dots

L e t S = 1 + s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty \dots (i)

\sin x \cdot S = s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty \dots (i i)

(i) - (i i) :

S (1 - \sin x) = 1

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

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

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

= \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 12}{16 - 12}

= \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2}

x = 60 °

Answered by Rasheed.Sindhi last updated on 09/Sep/22

≪ AnOther Approach \dots ≫

1 + s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty = 4 + 2 \sqrt{3}, x = ?

L e t S = 1 + s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty

S = 1 + s i n x (1 + s i n x + {s i n}^{2} x + {s i n}^{3} x + \dots \dots . . \infty

S = 1 + s i n x . S

S - s i n x . S = 1

S = \frac{1}{1 - s i n x} = 4 + 2 \sqrt{3}

1 - s i n x = \frac{1}{4 + 2 \sqrt{3}} \cdot \frac{4 - 2 \sqrt{3}}{4 - 2 \sqrt{3}}

s i n x = 1 - \frac{4 - 2 \sqrt{3}}{16 - 12} = \frac{4 - 4 + 2 \sqrt{3}}{4}

s i n x = \frac{\sqrt{3}}{2}

x = 60 °

Commented by peter frank last updated on 09/Sep/22

thanks

Commented by Tawa11 last updated on 15/Sep/22

Great sir .

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