Why do the solutions to the equation cos²x-3/4 lie in all four quadrants? Please explain why.

Expert Answers
hala718 eNotes educator| Certified Educator

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.