Question

Solve the following equations:
Find all value of a for which the equation $cos2x+asinx=2a-7$ possess a solution.

Answer 1

$cos2x+asinx=2a-7$

$1-2si{n}^{2}x+asinx=2a-7$

$sinx=u$

$1-2u^2+au=2a-7$

$\Rightarrow -2u^2+au=2a-8$

$\Rightarrow 2u^2-au+(2a-8)=0$

$\Rightarrow u=\frac{a\pm \sqrt{a^2+8(2a-8)}}{4}$

$D⩾0\Rightarrow a^2+8(2a-8)⩾0$

$\Rightarrow a^2+16a-64⩾0$

$\Rightarrow a^2+16a+64⩾128$

$\Rightarrow (a+8{)}^2⩾128$

$\Rightarrow -\sqrt{128}\le a+8\le \sqrt{128}$

$-\sqrt{128}-8\le a\le \sqrt{128}-8$

$-8(\sqrt{2}+1)\le a\le 8(\sqrt{2}-1)...\left(1\right)$

$\to \alpha +\beta =a/2,\alpha \beta =\frac{2a-8}{2}$

$\alpha ,\beta =sinxϵ[-1,1]$

$\Rightarrow \alpha +\beta ϵ[-2,2],\alpha \beta ϵ[-1,1]$

$\Rightarrow -2\le \frac{a}{2}\le 2$

$-4\le a\le 4...\left(2\right)$

$\frac{2a-8}{2}ϵ[-1,1]$

$-1\le a-4\le 1$

$3\le a\le 5...\left(3\right)$

from (1) ,(2) & (3)

$-8(\sqrt{2}+1)\le a\le 8(\sqrt{2}-1)$

$-4\le a\le 4$

$3\le a\le 5$

$\Rightarrow 3\le a\le 8(\sqrt{2}-1)$