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.