If `sintheta=1/2` then `theta=`

Options

A. `npi+(-1)^npi/6, n in Z`
B. `npi+(-1)^npi/3, n in Z`
C. `npipmpi/6, n in Z`
D. `npipmpi/3, n in Z`

Explanation

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

### Step 1: Identify the angle
The sine function equals \( \frac{1}{2} \) at specific angles. We know from trigonometric values that:

\[ \sin 30^\circ = \frac{1}{2} \]

### Step 2: Find all possible angles
The sine function is positive in the first and second quadrants. Therefore, the general solutions for \( \theta \) can be expressed as:

\[ \theta = 30^\circ + 360^\circ n \quad \text{(for the first quadrant)} \]
\[ \theta = 180^\circ - 30^\circ + 360^\circ n = 150^\circ + 360^\circ n \quad \text{(for the second quadrant)} \]

where \( n \) is any integer.

### Step 3: Write the final solution
Thus, the complete set of solutions for \( \theta \) is:

\[ \theta = 30^\circ + 360^\circ n \quad \text{or} \quad \theta = 150^\circ + 360^\circ n \quad \text{for } n \in \mathbb{Z} \]

### Summary of the solution:
- The angles where \( \sin \theta = \frac{1}{2} \) are \( 30^\circ \) and \( 150^\circ \), with periodicity of \( 360^\circ \).

---