Find ALL solutions of the interval [0, 2pi)

Question

Find ALL solutions of the interval [0, 2π)

cos 2x = √3/2

If there are some potential answers that won't fit, please say the reason.

Thank you

Answer 1

I need to solve cos(2x) = √3/2 for x ∈ [0, 2π).

Step 1: Find where cos(θ) = √3/2

First, let me identify where the cosine equals √3/2.

cos(θ) = √3/2 when:

1. θ = π/6 (30°)
2. θ = 11π/6 (330°)

These are in Quadrants I and IV where cosine is positive.

Step 2: Set up equations with the substitution

Since we have cos(2x) = √3/2, let θ = 2x:

2x = π/6 + 2πk or 2x = 11π/6 + 2πk (where k is any integer)

Step 3: Solve for x

Divide by 2:

x = π/12 + πk or x = 11π/12 + πk

Step 4: Find all solutions in [0, 2π)

Now I need to find which values of k give x in [0, 2π).

From x = π/12 + πk:

1. k = 0: x = π/12 ✓
2. k = 1: x = π/12 + π = 13π/12 ✓
3. k = 2: x = π/12 + 2π = 25π/12 ≈ 6.545 > 2π ✗ (too large)
4. k = -1: x = π/12 - π = -11π/12 < 0 ✗ (negative)

From x = 11π/12 + πk:

1. k = 0: x = 11π/12 ✓
2. k = 1: x = 11π/12 + π = 23π/12 ✓
3. k = 2: x = 11π/12 + 2π = 35π/12 ≈ 9.163 > 2π ✗ (too large)
4. k = -1: x = 11π/12 - π = -π/12 < 0 ✗ (negative)

Final Answer

The four solutions in [0, 2π) are:

x=π12,11π12,13π12,23π12x=12π​,1211π​,1213π​,1223π​

Or in decimal form: x≈0.262,2.880,3.403,6.021 radiansx≈0.262,2.880,3.403,6.021 radians

Why some potential answers don't fit:

1. Values with k = 2 give x ≥ 2π, which is outside our interval [0, 2π)
2. Values with k = -1 give negative x values, which are below 0

Answer 2

cos2x = (sqr3)/2

2x=arccos((sqr3)/2) = pi/6, 11pi/6, 13pi/6, 23pi/6, ...

x = pi/12, 11pi/12, 13pi/12, 23pi/12

= 15, 165, 195, 345 degrees

but cos2x > 0

cos30 >0, cos330>0, cos390>0, cos690=cos330>0

Answer 3

2x = cos^-1(√3/2)

2x = -π/6 + 2kπ or π/6 + 2kπ, k any integer

x = -π/12 + kπ or π/12 + kπ, k any integer

k=0: -π/12 , π/12

k=1: 11π/12, 13π/12

k=2: 23π/12, 25π/12

The lined-out solutions are outside the target interval.

Check out solutions on Desmos:

desmos.com/calculator/9guwipwqhd

Note: the only rejected solutions are those outside [0, 2π).