Question

Хелп 35 баллов 3sinx+14cosx

Answer 1

$3\, sinx+14\, cosx=\sqrt{205}\\\\3^2+14^2=9+196=205\\\\\\3\, sinx+14\, cosx=\sqrt{205}\, \Big|:\sqrt{205}\\\\\frac{3}{\sqrt{205}}\, sinx+\frac{14}{\sqrt{205}}\, cosx=1\\\\(\frac{3}{\sqrt{205}})^2+(\frac{14}{\sqrt{205}})^2=1\; \; \Rightarrow \; \; cosa=\frac{3}{\sqrt{205}}\; ,\; \; sina=\frac{14}{\sqrt{205}}\\\\cosa\cdot sinx+sina\cdot cosx=1\; \; ,\; \; a=(-1)^{n}arcsin\frac{14}{\sqrt{205}}+2\pi n\\\\sin(a+x)=1\\\\a+x=\frac{\pi}{2}+2\pi k\; ,\; k\in Z\\\\x=-a+\frac{\pi}{2}+2\pi k\; ,\; k\in Z$

$x=-(-1)^{n}arcsin\frac{14}{\sqrt{205}}+\frac{\pi}{2}+2\pi (k-n)\; ,\\\\x=(-1)^{n+1}arcsin\frac{14}{\sqrt{205}}+\frac{\pi}{2}+2\pi m\; ,\; m\in Z$

Answer 2

3sinx+14cosx=√205

(3/√205)²+(14/√205)²=(9+196)/205=1⇒3/√205=cosβ;   14/√205=sinβ

β=arcsin(14/√205)

sin(x+β)=1

x=-β+π/2+2πn; n∈Z

х=-1*arcsin(14/√205)+π/2+2πn; n∈Z