Question

рншить уравнение
sin3x + cos3x=(корень из 2)sin x

Answer 1

sin3x+sin(π/2-3x)=√2sinx
2sinπ/4cos(3x-π/4)=√2sinx
√2cos(3x-π/4)-√2cos(π/2-x)=0
√2(cos(3x-π/4)-cos(π/2-x))=0
cos(3x-π/4)-cos(π/2-x)=0
-2sin(x+π/4)sin(2x-3π/4)=0
sin(x+π/4)=0⇒x+π/4=πn⇒x=-π/4+πn
sin(2x-3π/4)=0⇒2x-3π/4=πn⇒2x=3π/4+πn⇒x=3π/8+πn/2

Answer 2

$1)\; \; sin3x+cos3x=sin3x+sin(\frac{\pi}{2}-x)=\\\\=2sin\frac{3x+\frac{\pi}{2}-3x}{2}\cdot cos\frac{3x-\frac{\pi}{2}+3x}{2}=2sin(\frac{\pi}{4})\cdot cos(3x-\frac{\pi}{4})=\\\\=\sqrt2\cdot cos(3x-\frac{\pi}{4})\\\\\\2)\; \; \sqrt2sinx=\sqrt2cos(\frac{\pi}{2}-x)\\\\3)\; \; \sqrt2cos(3x-\frac{\pi}{4})=\sqrt2cos(\frac{\pi}{2}-x)\\\\cos(3x-\frac{\pi}{4})-cos(\frac{\pi}{2}-x)=0\\\\-2sin(x+\frac{\pi}{8})sin(2x-\frac{3\pi}{8})=0\\\\a)\; \; sin(x+\frac{\pi}{8})=0$

$x+\frac{\pi}{8}=\pi n,\; n\in Z\\\\x=-\frac{\pi}{8}+\pi n,\; n\in Z\\\\b)\; \; sin(2x-\frac{3\pi}{8})=0\\\\2x-\frac{3\pi}{8}=\pi k,\; k\in Z\\\\2x=\frac{3\pi}{8}+\pi k,\; k\in Z\\\\x=\frac{3\pi}{16}+\frac{\pi k}{2},\; k\in Z$