Question

What is the mapping rule for a 45-degree rotation?

a) (x' = x cos(45) - y sin(45)), (y' = x sin(45) + y cos(45))

b) (x' = x sin(45) + y cos(45)), (y' = y sin(45) - x cos(45))

c) (x' = x cos(45) + y sin(45)), (y' = y cos(45) - x sin(45))

d) (x' = x sin(45) - y cos(45)), (y' = x cos(45) + y sin(45))

Answer 1

The correct mapping rule for a 45-degree rotation is (a) (x' = x cos(45) - y sin(45)), (y' = x sin(45) + y cos(45)), involving trigonometric functions for the counter-clockwise rotation of points in the coordinate plane.

Step-by-step explanation:

The mapping rule for a 45-degree rotation about the origin can be represented as (x' = x cos(45) - y sin(45)), (y' = x sin(45) + y cos(45)). This corresponds to choice (a) in the provided options. When performing this transformation, the original coordinates (x, y) are rotated counter-clockwise by 45 degrees to result in the new coordinates (x', y'). The cosine and sine of 45 degrees are involved in this transformation because rotation in the coordinate plane involves both the x and y components changing in a consistent manner according to trigonometric functions, specifically as a function of the rotation angle.