The most general value of `theta` which satisfies both the equation in If `sin theta = 1 //2 and cos theta = sqrt3 //2, `then the general value of `theta` which satisfies both the equation is

Options

A. `2 n pi + pi/3`
B. `2n pi + pi/4`
C. `2 n pi + pi/6`
D. none

Answer

To find the most general value of \( \theta \) that satisfies both equations \( \sin \theta = \frac{1}{2} \) and \( \cos \theta = \frac{\sqrt{3}}{2} \), we can follow these steps:

### Step 1: Identify the angles for \( \sin \theta = \frac{1}{2} \)
The sine function equals \( \frac{1}{2} \) at specific angles. From the unit circle, we know:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2n\pi \]
where \( n \) is any integer.

### Step 2: Identify the angles for \( \cos \theta = \frac{\sqrt{3}}{2} \)
Similarly, the cosine function equals \( \frac{\sqrt{3}}{2} \) at:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{11\pi}{6} + 2n\pi \]

### Step 3: Find common angles
Now, we need to find the common angles from both sets:
1. From \( \sin \theta = \frac{1}{2} \):
- \( \frac{\pi}{6} + 2n\pi \)
- \( \frac{5\pi}{6} + 2n\pi \)

2. From \( \cos \theta = \frac{\sqrt{3}}{2} \):
- \( \frac{\pi}{6} + 2n\pi \)
- \( \frac{11\pi}{6} + 2n\pi \)

The common angle is:
\[ \theta = \frac{\pi}{6} + 2n\pi \]

### Step 4: Write the general solution
Thus, the most general solution for \( \theta \) that satisfies both equations is:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{for any integer } n \]