`sin^2 theta = 1 //4`, then `theta `=

Options

A. `n pi pm pi//6`
B. `n pi + pi//3`
C. `n pi pm pi//6`
D. `n pi`

Answer

To solve the equation \( \sin^2 \theta = \frac{1}{4} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^2 \theta = \frac{1}{4} \]
We can express \( \frac{1}{4} \) as \( \left(\frac{1}{2}\right)^2 \):
\[ \sin^2 \theta = \left(\frac{1}{2}\right)^2 \]

### Step 2: Take the square root
Taking the square root of both sides, we get:
\[ \sin \theta = \frac{1}{2} \quad \text{or} \quad \sin \theta = -\frac{1}{2} \]

### Step 3: Find the angles for \( \sin \theta = \frac{1}{2} \)
The sine function equals \( \frac{1}{2} \) at specific angles:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{(where \( n \) is any integer)} \]

### Step 4: Find the angles for \( \sin \theta = -\frac{1}{2} \)
The sine function equals \( -\frac{1}{2} \) at:
\[ \theta = \frac{7\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{11\pi}{6} + 2n\pi \quad \text{(where \( n \) is any integer)} \]

### Step 5: Combine the solutions
Thus, the general solutions for \( \theta \) are:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad \theta = \frac{7\pi}{6} + 2n\pi \quad \text{or} \quad \theta = \frac{11\pi}{6} + 2n\pi \]

### Final Answer
The complete solution set is:
\[ \theta = n\pi \pm \frac{\pi}{6} \quad \text{for any integer } n \]

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