Question

if 0<=x<π/2 then what is the value of x in sin x-sin 2x + sin 3x=0?​

Answer 1

Answer:

x=0 or x=π/3

Step-by-step explanation:

Given, 0≤x<π/2,

so,

$\sin(x) - \sin(2x) + \sin(3x) = 0$

$= > \sin(x) + \sin(3x) - \sin(2x) = 0$

We know that, sin(c)+sin(d)=2sin((c+d)/2)cos((c-d)/2)

so,

$= > 2 \sin( \frac{x + 3x}{2} ) \cos( \frac{x - 3x}{2} ) - \sin(2x) = 0$

$= > 2 \sin(2x) \cos( - x) - \sin(2x) = 0$

we know, cos(-a)=cos(a),

so,

$= > 2 \sin(2x) \cos(x) - \sin(2x) = 0$

$= > \sin(2x) (2 \cos(x) - 1) = 0$

$either \: \: \sin(2x) = 0 \: \: or \: \: (2 \cos(x) - 1) = 0$

$either \: \: 2x = 0 \: \: or \: \: \cos(x) = \frac{1}{2}$

$either \: \: x = 0 \: \: or \: \: x = \frac{\pi}{3}$

as, we know, sin(0)=0 and cos(π/3)=1/2