The-equation-asinx-cos2x-2a-7-possesses-a-solution-if-1-a-gt-6-2-2-a-6-3-a-gt-2-4-a-

Question Number 15263 by Tinkutara last updated on 08/Jun/17

The equation a \sin x + \cos 2 x = 2 a - 7

possesses a solution if

(1) a > 6

(2) 2 ⩽ a ⩽ 6

(3) a > 2

(4) a

Answered by ajfour last updated on 08/Jun/17

a \sin x + 1 - 2 \sin^{2} x = 2 a - 7

\sin^{2} x - \frac{a}{2} \sin x = 4 - a

{(\sin x - \frac{a}{4})}^{2} = \frac{a^{2}}{16} - a + 4

{(\sin x - \frac{a}{4})}^{2} = {(\frac{a}{4} - 2)}^{2}

∣ \sin x - \frac{a}{4} ∣=∣ \frac{a}{4} - 2 ∣

\Rightarrow \frac{a}{4} - 2 = \pm ∣ \sin x - \frac{a}{4} ∣

w i t h t h e - v e s i g n, a b o v e e q n . \Rightarrow

n o v a l u e o f a t o f u l f i l l t h e c o n d i t i o n;

w i t h t h e + v e s i g n i t i m p l i e s

\frac{a}{2} = 2 + \sin x

o r a = 4 + 2 \sin x

A s - 1 ⩽ \sin x ⩽ 1

w e c a n c o n c l u d e t h a t

2 ⩽ a ⩽ 6 .

Commented by Tinkutara last updated on 09/Jun/17

Thanks Sir!