given a point on a line segment (-4,5), what is the new point when the point is rotated clockwise about the origin 270 degrees?
To find the new point after rotating a point in the plane, we can use the rotation formulas.
Let's denote the original point as (x, y) = (-4, 5).
To rotate a point 270 degrees clockwise about the origin, we can use the following formula:
x' = x * cos(theta) + y * sin(theta)
y' = -x * sin(theta) + y * cos(theta)
where theta is the angle of rotation in radians.
Plugging in the values for x, y, and theta, we have:
x' = (-4) * cos(270°) + 5 * sin(270°)
y' = (-4) * sin(270°) + 5 * cos(270°)
To convert the angle from degrees to radians, we use the conversion factor: π/180
x' = (-4) * cos(270° * π/180) + 5 * sin(270° * π/180)
y' = (-4) * sin(270° * π/180) + 5 * cos(270° * π/180)
Now evaluating the trigonometric functions:
x' = (-4) * cos(3π/2) + 5 * sin(3π/2)
y' = (-4) * sin(3π/2) + 5 * cos(3π/2)
cos(3π/2) = 0 and sin(3π/2) = -1, so:
x' = (-4) * 0 + 5 * (-1)
y' = (-4) * (-1) + 5 * 0
Thus, the new point after rotating (-4, 5) 270 degrees clockwise about the origin is:
(x', y') = (0, -4)