https://file.aiquickdraw.com/mathbot/file/1725553353478z0n8v72j.jpg

The problem in the image asks for the coordinates of a point \((3, 5)\) after the coordinate system has been rotated by \(\theta = 30^\circ\). To solve this, we need to apply the 2D rotation formula for rotating a point in the plane.

### Rotation Formula:
Given a point \((x, y)\) and an angle \(\theta\), the coordinates of the rotated point \((x', y')\) are given by:

\[
x' = x \cdot \cos(\theta) - y \cdot \sin(\theta)
\]
\[
y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)
\]

### Step-by-step solution:
1. The point is \((x, y) = (3, 5)\).
2. The angle of rotation is \(\theta = 30^\circ\).
3. From trigonometric tables, we know:
   \[
   \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2}
   \]

4. Apply the formulas for the new coordinates:

\[
x' = 3 \cdot \frac{\sqrt{3}}{2} - 5 \cdot \frac{1}{2}
\]
\[
y' = 3 \cdot \frac{1}{2} + 5 \cdot \frac{\sqrt{3}}{2}
\]

Let me now calculate these values to determine the correct coordinates.After performing the rotation of the point \((3, 5)\) by \(30^\circ\), the new coordinates are approximately:

\[
x' \approx 0.098, \quad y' \approx 5.830
\]

Looking at the options provided in your image, it appears that the correct choice would be:

**Option b** \(\left( \frac{3\sqrt{3} - 5}{2}, \frac{3 + 5\sqrt{3}}{2} \right)\).

Would you like further explanation or details on this calculation? Here are some related questions:

1. How is the 2D rotation formula derived?
2. How do rotations affect the orientation of vectors?
3. What are the real-world applications of 2D coordinate rotation?
4. How does the cosine and sine of an angle affect the new coordinates after rotation?
5. Can the rotation formula be applied to three-dimensional space?

**Tip:** When performing geometric transformations, especially rotations, always consider the unit circle to better understand how angles correspond to sine and cosine values.

The xy-coordinate system has been rotated θ degrees from the xy-coordinate system. The coordinates of a point in the xy-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. θ = 30°, (3, 5).

Learn how to find the new coordinates of the point (3, 5) after rotating the xy-coordinate system by 30°. This problem covers key concepts in coordinate geometry and trigonometry, using the 2D rotation formula for transforming points.

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