Given a point on a line segment (−4, 5), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?
To rotate a point counterclockwise around the origin, we can use the following rotation formula:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Where (x, y) represents the original point coordinates, (x', y') represents the rotated point coordinates, and θ represents the angle of rotation.
Given the original point (-4, 5) and a rotation of 270 degrees counterclockwise, we have:
x = -4
y = 5
θ = 270 degrees
Using the rotation formula, we can calculate the coordinates of the rotated point:
x' = -4 * cos(270°) - 5 * sin(270°)
y' = -4 * sin(270°) + 5 * cos(270°)
To calculate the trigonometric functions of 270 degrees, we can use the fact that cos(270°) = 0 and sin(270°) = -1. Substituting these values into the rotation formula:
x' = -4 * 0 - 5 * (-1) = 0 - (-5) = 5
y' = -4 * (-1) + 5 * 0 = 4 + 0 = 4
Therefore, the new point after rotating the original point (-4, 5) counterclockwise about the origin 270 degrees is (5, 4).