Find the sum of the three smallest positive values of θ such that 4 cos^2(20-π)=3?

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Answer 1 · 2018-05-07T03:24:09

#(13pi)/12#

I will assume it says:

#4 cos^2(2theta-π)=3#

Let #u# be #2theta-pi#

#4cos^2u=3#

#cos^2u= 3/4#

#cosu= +-sqrt3/2#

Now I will list all my solutions from #(-pi, pi)# instead of #(0, 2pi)# for a reason I will discuss later:

#u= -(5pi)/6, -(pi)/6, pi/6, (5pi)/6#

Now replace #u# with #2theta-pi#:

The reason why I listed solutions from #-pi# is because I will be adding #pi# here:
#2theta-pi=-(5pi)/6, -(pi)/6, pi/6, (5pi)/6#

#2theta= pi/6, (5pi)/6, (7pi)/6, (11pi)/6#

#theta= pi/12, (5pi)/12, (7pi)/12, (11pi)/12#

Sum of the three smallest positive solutions:
#pi/12+(5pi)/12+(7pi)/12= (13pi)/12#

graph{4(cos(2x-pi))^2-3 [-10, 10, -5, 5]}