For the equation 4 cos theta - sqrt12 =0, where 0< theta < or equal 2pi determine the measure of theta. One answer I got was 2pi/3, 4pi/3 and another time it totally different 5 pi/6, 7pi/6. Are either of these right if not HELP!

To solve the equation 4 cos θ - √12 = 0 and find the values of θ, we can follow these steps:

Step 1: Simplify the equation
Start by isolating the cosine term by adding √12 to both sides of the equation:
4 cos θ = √12

Step 2: Solve for cos θ
Divide both sides of the equation by 4:
cos θ = √12 / 4

Step 3: Simplify the right side of the equation
√12 / 4 is equivalent to (√4 * √3) / 4
Since √4 = 2, we can rewrite the equation as:
cos θ = 2√3 / 4

Step 4: Simplify further
Divide the numerator and denominator by 2:
cos θ = √3 / 2

Step 5: Determine possible values of θ
Using the unit circle or trigonometric ratios, we know that cos θ = √3 / 2 has two possible solutions: π/6 and 11π/6.

To confirm these solutions, we can substitute them back into the original equation:
For θ = π/6:
4 cos(π/6) - √12 = 0
4 (√3/2) - √12 = 0
2√3 - √12 = 0
2√3 - 2√3 = 0
0 = 0 (true)

For θ = 11π/6:
4 cos(11π/6) - √12 = 0
4 (√3/2) - √12 = 0
2√3 - √12 = 0
2√3 - 2√3 = 0
0 = 0 (true)

Therefore, both θ = π/6 and θ = 11π/6 are correct solutions for the given equation.