Question 1160202
Find all solutions of the equation on the interval [{{{0}}},{{{2pi}}})

{{{-4sin(x) = -cos^2(x)+4}}}


{{{cos^2(x)-4sin(x) -4= 0}}}............use the following identity : {{{cos^2 (x )=1-sin^2(x)}}}

{{{1-sin^2(x)-4sin(x) -4= 0}}}

{{{-sin^2(x)-4sin(x) -3= 0}}}......both sides multiply by {{{-1}}}

{{{sin^2(x)+4sin(x) +3= 0}}}.........let{{{ sin(x)=u}}}

{{{u^2+4u +3= 0}}}.......factor

{{{u^2+u+3u +3= 0}}}

{{{(u^2+u)+(3u +3)= 0}}}

{{{u(u+1)+3(u +1)= 0}}}

{{{(u + 1) (u + 3) = 0}}}

solutions:

{{{u=-1}}}

{{{u=-3}}}

substitute {{{u}}} back

{{{sin(x)=-1}}}-> general solutions for {{{sin (x )=-1}}}: {{{x=3pi/2+2pi*n}}}

for interval {{{0<=x<2pi}}}  will be => {{{highlight(x=3pi/2)}}}

{{{sin(x)=-3}}}-> {{{sin(x)}}} can't be smaller than {{{-1}}} for real solutions => no solution exist


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