Question

Solve the following equations.
$asin\frac{x}{2}-(sinx+sin\frac{3x}{2})=0.$

Answer 1

$asin\frac{x}{2}-(sinx+sin\frac{3x}{2})=0.$

$asin\frac{x}{2}-[sinx+sin(x+\frac{x}{2})]=0[\because x+\frac{x}{2}=\frac{3}{2}x]$

we know that $sin(a+b)=sinacosb+cosasinb$

$sin2a=2sinacosa$

$\Rightarrow asinx/2-[2sinx/cosx/2+sinxcosx/2+cosxsinx/2]=0$

$\Rightarrow asinx/2-2cosx/2sinx/2+2cosx/2cosx/2+cosxsinx/2=0$

$\{\because sin2a=2sinacosa\}$

$\Rightarrow asinx/2-2cosx/2sinx/2+2co{s}^{2}x/2sinx/2+cosxsinx/2=0$

$\Rightarrow sinx/2[a-2cosx/2+2co{s}^{2}x/2+cosx]=0$

$\Rightarrow sinx/2=0$ or $a-2cosx/2+2co{s}^{2}x/2+cosx=0$

$\Rightarrow \frac{x}{2}=si{n}^{-1}\left(0\right)$ or $a=2cosx/2+2co{s}^{2}x/2+cosx$

$\Rightarrow \frac{x}{2}=0$ or $\because cos2A=2co{s}^{2}A-1$

$\Rightarrow x=0$ or $a=2cosx/2+2co{s}^{2}x/2+2co{s}^{2}x/2-1$

$a=4co{s}^{2}x/2+2cosx/2-1$