If 1+sin x+sin ^{2} x+ldots up to infty = 4+2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The given expression is an infinite geometric series with first term $a = 1$ and common ratio $r = \sin x$. Since the sum is $4+2 \sqrt{3}$,we use

Vedclass · Accepted Answer

$\frac{\pi}{3}, \frac{2 \pi}{3}$

Vedclass · Answer

(C) The given expression is an infinite geometric series with first term $a = 1$ and common ratio $r = \sin x$. Since the sum is $4+2 \sqrt{3}$,we use the formula $S = \frac{a}{1-r}$: $\frac{1}{1-\sin x} = 4+2 \sqrt{3}$ $1-\sin x = \frac{1}{4+2 \sqrt{3}}$ Rationalizing the denominator: $1-\sin x = \frac{4-2 \sqrt{3}}{(4+2 \sqrt{3})(4-2 \sqrt{3})} = \frac{4-2 \sqrt{3}}{16-12} = \frac{4-2 \sqrt{3}}{4} = 1 - \frac{\sqrt{3}}{2}$ Thus,$\sin x = \frac{\sqrt{3}}{2}$. For $0 < x < \pi$,the values of $x$ satisfying $\sin x = \frac{\sqrt{3}}{2}$ are $x = \frac{\pi}{3}$ and $x = \pi - \frac{\pi}{3} = \frac{2 \pi}{3}$.

If $1+\sin x+\sin ^{2} x+\ldots$ up to $\infty = 4+2 \sqrt{3}$,where $0 < x < \pi$ and $x \neq \frac{\pi}{2}$,then $x$ is equal to

Similar Questions

In an increasing geometric progression of positive terms,the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is $49$. Then the sum of the $4^{\text{th}}$,$6^{\text{th}}$,and $8^{\text{th}}$ terms is:

Let $a, b, c, d$ and $p$ be any non-zero distinct real numbers such that $(a^{2}+b^{2}+c^{2}) p^{2} - 2(ab+bc+cd) p + (b^{2}+c^{2}+d^{2}) = 0$. Then:

The sum of some terms of a geometric progression is $728$. If the common ratio is $3$ and the last term is $486$,what is the first term of the progression?

Let $S_1$ be the sum of areas of the squares whose sides are parallel to the coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then,$\frac{S_1}{S_2}$ is equal to

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module