someone please help! how do you solve this...4cos^2x -3 =0?

Question

4cos^2x -3 =0

Answer 1

4cos^2(x) - 3 = 0

Bring the 3 over to the right hand side.

4cos^2(x) = 3

Divide both sides by 4.

cos^2(x) = 3/4

Now, here is the crucial part; take the square root of both sides. Note that whenever you take the square root of both sides, you have to have a "plus or minus" (which I'll denote +/-).

cos(x) = +/- sqrt(3/4)

Note that the square root of 3/4 means we take the square root of the top and bottom. sqrt(3/4) is the same as sqrt(3)/2

cos(x) = +/- sqrt(3) / 2

Thus, we have two equations to find solutions for:
cos(x) = sqrt(3)/2 and cos(x) = -sqrt(3)/2

Let's assume that your solutions are in the interval [0, 2pi).

cos(x) = sqrt(3)/2
Cosine is equal to sqrt(3)/2 at the points pi/6 and 11pi/6. Therefore, x = {pi/6, 11pi/6}

cos(x) = -sqrt(3)/2
Cosine is equal to -sqrt(3)/2 at the points 5pi/6 and 7pi/6.
Therefore, x = {5pi/6, 7pi/6}

Combining our solutions,

x = {pi/6, 11pi/6, 5pi/6, 7pi/6}