Question

For 0 ≤ x < 2π, which of the following solves 2cos(x/2) - 1 = 0?

Option 1: {4π/3}
Option 2: {5π/3}
Option 3: {π/3}
Option 4: {2π/3}

Answer 1

The solutions to the equation 2cos(x/2) - 1 = 0 for 0 ≤ x < 2π are {2π/3} and {5π/3}.

Step-by-step explanation:

To solve the equation 2cos(x/2) - 1 = 0, we can isolate the cosine term and solve for x.

1. Add 1 to both sides of the equation: 2cos(x/2) = 1

2. Divide both sides of the equation by 2: cos(x/2) = 1/2

3. Take the inverse cosine (cos^-1) of both sides to get the value of x/2. Remember to consider the range of x: x/2 = π/3 or x/2 = 5π/3

4. Multiply both sides of the equations by 2 to get the values of x: x = 2π/3 or x = 5π/3

So, the solutions to the equation 2cos(x/2) - 1 = 0 for 0 ≤ x < 2π are {2π/3} and {5π/3}.