In the coordinate plane, what is the rule for a $90°$ clockwise rotation about the origin that maps a point $(x,y)$ to its image?

• A. $(x,y)$ maps to $(-x,-y)$
• B. $(x,y)$ maps to $(y,-x)$
• C. $(x,y)$ maps to $(x,-y)$
• D. $(x,y)$ maps to $(-y,x)$

Answer: (B)

Verified step by step guidance

1. Recall that a rotation about the origin by an angle \( \theta \) transforms a point \( (x, y) \) to a new point \( (x', y') \) using the formulas: \[ x' = x \cos(\theta) - y \sin(\theta), \quad y' = x \sin(\theta) + y \cos(\theta) \].
2. For a 90-degree clockwise rotation, note that clockwise rotation by 90 degrees is equivalent to counterclockwise rotation by -90 degrees, so set \( \theta = -90^\circ \).
3. Substitute \( \theta = -90^\circ \) into the rotation formulas. Use the trigonometric values \( \cos(-90^\circ) = 0 \) and \( \sin(-90^\circ) = -1 \).
4. Apply these values to get: \[ x' = x \cdot 0 - y \cdot (-1) = y, \quad y' = x \cdot (-1) + y \cdot 0 = -x \].
5. Therefore, the rule for a 90-degree clockwise rotation about the origin is: \[ (x, y) \mapsto (y, -x) \].