If `theta` is an angle in standard position and its terminal side passes thorugh the point `(sqrt(3)/(2), -(1)/(2))` on the unit circle, then a possible value of `theta` is

Options

A. `(7pi)/(6)`
B. `(4pi)/(3)`
C. `(5pi)/(3)`
D. `(11pi)/(6)`

Answer

To find the angle \( \theta \) in standard position whose terminal side passes through the point \( \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \) on the unit circle, we can follow these steps:

### Step 1: Identify the coordinates
The point given is \( \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \). Here, the x-coordinate is \( \frac{\sqrt{3}}{2} \) and the y-coordinate is \( -\frac{1}{2} \).

### Step 2: Determine the quadrant
Since the x-coordinate is positive and the y-coordinate is negative, the point lies in the fourth quadrant.

### Step 3: Use trigonometric identities
In the unit circle, we know that:
- \( \cos(\theta) = x \)
- \( \sin(\theta) = y \)

Thus, we have:
- \( \cos(\theta) = \frac{\sqrt{3}}{2} \)
- \( \sin(\theta) = -\frac{1}{2} \)

### Step 4: Find the reference angle
The cosine value \( \frac{\sqrt{3}}{2} \) corresponds to an angle of \( \theta = \frac{\pi}{6} \) in the first quadrant.

### Step 5: Determine the angle in the fourth quadrant
Since we are in the fourth quadrant, we can find \( \theta \) using the formula:
\[ \theta = 2\pi - \text{reference angle} \]
Substituting the reference angle:
\[ \theta = 2\pi - \frac{\pi}{6} \]

### Step 6: Simplify the angle
To simplify:
\[ \theta = 2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6} \]

### Conclusion
Thus, a possible value of \( \theta \) is:
\[ \theta = \frac{11\pi}{6} \]

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