Given-4cos-2-x-sin-x-2sin-2-x-3sin-x-where-pi-2-x-pi-2-Find-the-sum-of-all-posibble-value-of-x-

Question Number 130047 by liberty last updated on 22/Jan/21

Given 4 \cos^{2} x \sin x - 2 \sin^{2} x = 3 \sin x

where - \frac{π}{2} ⩽ x ⩽ \frac{π}{2} . Find the sum of

all posibble value of x .

Answered by MJS_new last updated on 22/Jan/21

\sin x = s \Rightarrow \cos^{2} x = 1 - s^{2}

(s^{2} + \frac{s}{2} - \frac{1}{4}) s = 0

\Rightarrow s = 0 \lor s = - \frac{1}{4} \pm \frac{\sqrt{5}}{4}

\Rightarrow x = 0 \lor x = - \frac{3 π}{10} \lor x = \frac{π}{10}

\Rightarrow answer is - \frac{π}{5}

Answered by Alepro3 last updated on 22/Jan/21

\cos^{2} x = 1 - \sin^{2} x \Rightarrow

4 \sin x - 4 \sin^{3} x - 2 \sin^{2} x = 3 \sin x \Rightarrow

\sin x (4 \sin^{2} x + 2 \sin x - \sin x) = 0 \Rightarrow

o n e s o l u t i o n i s \sin x = 0 \Rightarrow x = k π

s o l v i n g t h e 2^{n d} d e g r e e e q u a t i o n w e h a v e

\sin x_{1, 2} = (- 2 \pm \sqrt{4 + 16}) / 8 = (- 1 \pm \sqrt{5}) / 4

s o x_{1, 2} = \sin^{- 1} [(- 1 \pm \sqrt{5}) / 4] + 2 k π a n d x_{3, 4} = π - \sin^{- 1} [[(1 - \sqrt{5}) / 4] + 2 k π \Rightarrow

t h e s u m o f a l l o f t h i s v a l u e o f x i s

Σ x = π + 5 k π

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