Why do the solutions to the equation cos²x-3/4 lie in all four quadrants? Please explain why.
I am assuming that you mean cos^2 x - 3/4 because 3.4 is not valid.
==> cos^2 x - 3/4 = 0
Let us solve the equation.
First add 3/4 to both sides.
==> cos^2 x = 3/4
Now we will take the root of both sides.
==> cos(x) = +-sqrt3/2
==> Then, we have two possible values for cosx.
Case(1): cosx =+ sqrt3/2 ==> x > 0. Then, x is in the first and fourth quadrants.
==> x1= pi/6 ( first quadrant)
==> x2= 2pi - pi/6 = 11pi/6 ( fourth quadrant).
Case(2): cosx = - sqrt3/2 ==> x < 0. Then, x is in the second and third quadrants.
==> x3= pi- pi/6 = 5pi/6 ( 2nd quadrant).
==> x4= pi + pi/6 = 7pi/6 ( 3rd quadrant).
Then, the solutions of cos^2 x - 3/4 are in all four quadrants.