Solve the following equation.

12 cosine squared x equals 312cos2x=3​,

0 less than or equals x less than 2 pi0≤x<2π

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Part 1

xx

​=

enter your response here

​(Type an exact​ answer, using

piπ

as needed. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.)

To solve the equation \( 12 \cos^2(x) = 3 \), we first rewrite the equation step by step:

### Step 1: Simplify the Equation
Divide both sides by 12 to isolate \( \cos^2(x) \):

\[
\cos^2(x) = \frac{3}{12}
\]

\[
\cos^2(x) = \frac{1}{4}
\]

### Step 2: Solve for \( \cos(x) \)
To solve for \( \cos(x) \), we take the square root of both sides. Remember that the square root can be positive or negative:

\[
\cos(x) = \pm \frac{1}{2}
\]

### Step 3: Find the General Solutions
We need to find the values of \( x \) in the range \( 0 \leq x < 2\pi \) for which \( \cos(x) = \frac{1}{2} \) or \( \cos(x) = -\frac{1}{2} \).

1. **For \( \cos(x) = \frac{1}{2} \):**
   - The solutions are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \).

2. **For \( \cos(x) = -\frac{1}{2} \):**
   - The solutions are \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).

### Step 4: List All Solutions
Combining all the solutions, we get:

\[
x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}
\]

These are the exact solutions within the interval \( 0 \leq x < 2\pi \).

### Final Answer
\[
x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}
\]

Would you like more details on any step or have any questions?  

### Related Questions:
1. How do you solve trigonometric equations involving sine or tangent?
2. What are the solutions for \( \cos(x) = \pm \frac{\sqrt{2}}{2} \) in the interval \( 0 \leq x < 2\pi \)?
3. How do you graph \( y = \cos(x) \) and identify specific values for \( x \)?
4. What are the double angle formulas for cosine and sine?
5. How can you use the unit circle to find cosine and sine values?

### Tip:
For trigonometric equations, always consider both positive and negative cases when taking the square root!

Solve the following equation: 12cos^2(x) = 3, 0 ≤ x < 2π

Discover how to solve the trigonometric equation 12cos^2(x) = 3 within the interval 0 ≤ x < 2π. This solution includes step-by-step instructions, key formulas, and relevant trigonometric concepts, suitable for high school students in Grades 10-12.

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