The x’y’-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.

a.Θ = 90o, (0, 3)

When a coordinate system is rotated by an angle \(\theta\), the coordinates \((x', y')\) in the rotated system can be found using the following transformation formulas:

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

Given:
- \(\theta = 90^\circ\)
- Coordinates in the original system: \((x, y) = (0, 3)\)

Now, apply the rotation:

1. **Compute \(x'\):**
   \[
   x' = x \cos 90^\circ + y \sin 90^\circ = 0 \times 0 + 3 \times 1 = 3
   \]

2. **Compute \(y'\):**
   \[
   y' = -x \sin 90^\circ + y \cos 90^\circ = -0 \times 1 + 3 \times 0 = 0
   \]

Thus, the coordinates in the rotated system are \((x', y') = (3, 0)\).

Would you like any further details or clarifications?

Here are 5 related questions to explore:
1. What is the general rotation formula for any angle \(\theta\)?
2. How does a negative angle of rotation affect the transformation?
3. What happens to the coordinates of a point after a \(180^\circ\) rotation?
4. How can you derive the inverse rotation transformation?
5. What is the impact of rotating by small angles, such as \(15^\circ\)?

**Tip:** When rotating by \(90^\circ\), \(\cos(90^\circ)\) becomes 0 and \(\sin(90^\circ)\) becomes 1, making the calculation particularly straightforward.

The x’y’-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 = 90°, (0, 3)

Learn how to find the coordinates of a point (0, 3) after a 90-degree rotation in a coordinate system. This problem involves key concepts in coordinate geometry, including rotation transformation formulas and theorems. Suitable for students in Grades 10-12.

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