Question

sinx.sin3x=1/2 then general solution of x=?​

Answer 1

Answer:

The general solution of x is $x=[\frac{(2n+1)\pi }{4}, \frac{(3n+1)\pi }{6}]$

Step-by-step explanation:

The trigonometric equation is: $sinx\times sin 3x = \frac{1}{2}$

The expansion of sin 3x is:

$sin3x=3sinx-4sin^{3}x$

Solve as follows:

$sinx\times sin 3x = \frac{1}{2}\\sinx(3sinx-4sin^{3}x)= \frac{1}{2}\\8sin^{4}x-6sin^{2}x+1=0$

The resultant equation is a quadratic equation. Solve as follows:

$8sin^{4}x-6sin^{2}x+1=0\\8sin^{4}x-4sin^{2}x-2sin^{2}x+1=0\\4sin^{2}x(2sin^{2}x-1)-1(2sin^{2}x-1)=0\\(2sin^{2}x-1)(4sin^{2}x-1)=0\\sin^{2}x=\frac{1}{2}\ or\ \frac{1}{4}\\ sinx=\pm\frac{1}{\sqrt{2} }\ or\ \pm\frac{1}{2}$

For $sinx=\pm\frac{1}{\sqrt{2}}$ the solution of x is:

$x=\frac{n\pi }{2}+\frac{\pi }{4} \\=\frac{(2n+1)\pi }{4}$

For $sinx=\pm\frac{1}{2}$ the solution of x is:

$x=\frac{n\pi }{2}+\frac{\pi }{6} \\=\frac{(3n+1)\pi }{6}$