How do yo solve #2Cos^2(x) - 2sen(x) + 12 = 0# ?

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Answer 1 · 2018-03-05T08:36:13

See explanation

We want to solve the equation

#2cos^2(x)-2sin(x)+12=0#

Use #cos^2(x)=1-sin^2(x)#

#2(1-sin^2(x))-2sin(x)+12=0#

Simplify

#-2sin^2(x)-2sin(x)+14=0#

#=>sin^2(x)+sin(x)-7=0#

Let #u=sin(x)# and solve the quadratic equation

#u^2+u-7=0#

#=>u=(-1+-sqrt(29))/2#

#=>sin(x)=(-1+-sqrt(29))/2#

So the equation have no solutions within the real numbers
because #-1<=sin(x)<=1#, if you want complex solutions you can always take the inverse sine