sin(3x) = cos(x + pi/4)
Thanks
$\displaystyle \sin(3x) = \cos( x + \frac{\pi}{4} ) $
$\displaystyle \sin(3x) = \sin( \frac{\pi}{2} - x - \frac{\pi}{4} )$
$\displaystyle \sin(3x) = \sin( \frac{\pi}{4} - x )$
$\displaystyle 3x = 2n\pi + \frac{\pi}{4} - x $
$\displaystyle x = \frac{(8n+1)\pi}{16} $
Or
$\displaystyle 3x = (2n+1)\pi - \frac{\pi}{4} + x $
$\displaystyle x = \frac{( 8n + 3 )\pi}{8}$
i have plotted this function, sin(3x) - cos(x + pi/4) = 0, for x = 0 to 2pi
it does have 6 roots for x = 0 to 2 pi
using the sum to product formula, we have this
-2 sin (pi/8 -2x) cos (x + pi/8) = 0
this one form does not generate the 6 roots as shown in the plot, i wonder