Question

А)sqrt3cos x + sin x=1 б)1/cos^2x+3/sin(p/2+x)+2=0 !

Answer 1

$1) \frac{ \sqrt{3} }{2} cosx+ \frac{1}{2} sinx= \frac{1}{2} \\ \\ cosxsin \frac{ \pi }{3} +sinxcos \frac{ \pi }{3} =\frac{1}{2} \\ \\ sin(x+\frac{ \pi }{3} )=(-1)^{k} \frac{ \pi }{6} + \pi k \\ \\ x=(-1)^{k} \frac{ \pi }{6} + \pi k- \frac{ \pi }{3}$

$2) \frac{1}{cos^{2} x} + \frac{3}{sin( \frac{ \pi }{2}+x) } +2=0 \\ \\ \frac{1}{cos^{2} x} + \frac{3}{cosx} =-2 \\ \\ \frac{1+3cosx}{cos^{2}x } = \frac{-2}{1} \\ \\ 1+3cosx=-2cos^{2} x \\ 1+3cosx=-2* \frac{1+cos2x}{2} \\ 1+3cosx=-1-cos2x \\ \\ 3cosx+cos2x=-2 \\ \\ 3cosx+2cos^{2} x-1=-2 \\ \\ 3cosx+2cos^{2} x=-1 \\ \\ cosx=t \\ \\ 3t+2t^{2} +1=0 \\ \\ d=9-8=1 \\ t1=(-3-1)/4=-1 \\ t2=(-3+1)/4=-2/4=-1/2 \\ \\ cosx=-1 \\ x= \pi +2 \pi k \\ \\ cosx=-1/2 \\ x=+/- \frac{2 \pi }{3} +2 \pi k$