Question

Решить тригонометрическое уравнение. 47 баллов!!1
$\sin(x) + \sqrt{ \cos(x) } = 0$
​

Answer 1

$\displaystyle ODZ: cosx\geq 0; x\in I; IV$

$\displaystyle \sqrt{cosx}\geq 0; sinx\leq 0; x\in IV$

$x^{2} \displaystyle \sqrt{cosx}= -sinx\\ cosx=(1-cos^2x)\\cos^2x+cosx-1=0\\\\D=1+4=5\\\\cosx=\frac{-1+\sqrt{5}}{2}<1\\\\cosx\neq \frac{-1-\sqrt{5}}{2}<-1$

$\displaystyle x= \pm arccos (\frac{\sqrt{5}-1}{2})+2\pi n; n\in Z\\\\ x\in IV\\x=- arccos (\frac{\sqrt{5}-1}{2})+2\pi n ; n\ inZ$