What are the coordinates of the image of the point (1, 0) on the terminal side of an angle after a clockwise rotation of `(pi)/(6)` radians?

Options

A. `((sqrt(3))/(2), -(1)/(2)))`
B. `((1)/(2), -(sqrt(3))/(2))`
C. `(-(sqrt(3))/(2), 1)`
D. `((1)/(2), -(1)/(2))`

Answer

To find the coordinates of the image of the point (1, 0) after a clockwise rotation of \(\frac{\pi}{6}\) radians, we can follow these steps:

### Step 1: Understand the Initial Point
The initial point given is (1, 0). This point lies on the x-axis, which corresponds to an angle of 0 radians.

### Step 2: Convert to Polar Coordinates
In polar coordinates, the point (1, 0) can be represented as:
- \( r = 1 \) (the distance from the origin)
- \( \theta = 0 \) (the angle with the positive x-axis)

### Step 3: Determine the New Angle
Since we are rotating the point clockwise by \(\frac{\pi}{6}\) radians, we need to subtract this angle from the original angle:
\[ \theta' = 0 - \frac{\pi}{6} = -\frac{\pi}{6} \]

### Step 4: Calculate the New Coordinates
Using the polar to rectangular coordinate conversion formulas:
- \( x = r \cos(\theta') \)
- \( y = r \sin(\theta') \)

Substituting \( r = 1 \) and \( \theta' = -\frac{\pi}{6} \):
\[ x = 1 \cdot \cos\left(-\frac{\pi}{6}\right) \]
\[ y = 1 \cdot \sin\left(-\frac{\pi}{6}\right) \]

### Step 5: Evaluate the Trigonometric Functions
Now we evaluate the cosine and sine:
- \(\cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
- \(\sin\left(-\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}\)

### Step 6: Substitute Back to Find Coordinates
Now substituting these values back:
\[ x = \frac{\sqrt{3}}{2} \]
\[ y = -\frac{1}{2} \]

### Final Coordinates
Thus, the coordinates of the image of the point (1, 0) after a clockwise rotation of \(\frac{\pi}{6}\) radians are:
\[ \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \]

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