Question

Given a point (x, y), what is the new point when the point is rotated counterclockwise about the origin 270 degrees

Answer 1

The new point, when the point (x, y) is rotated counterclockwise about the origin by 270 degrees, is (-y, x).

The step by step calculation:

Step 1: Calculate the angle of rotation in radians.

- 270 degrees is equivalent to 3/2π radians.

Step 2: Use the rotation formula for counterclockwise rotation about the origin:

- For a point (x, y) rotated counterclockwise by θ radians, the new point is (x*cos(θ) - y*sin(θ), x*sin(θ) + y*cos(θ)).

Step 3: Substitute the values:

- θ = 3/2π, x = x, and y = y.

- New x = x*cos(3/2π) - y*sin(3/2π) = x*0 - y*(-1) = y.

- New y = x*sin(3/2π) + y*cos(3/2π) = x*(-1) + y*0 = -x.

Step 4: The new point is (-y, -x).

So, when the point (x, y) is rotated counterclockwise about the origin by 270 degrees, the new point is (-y, -x).

Answer 2

To rotate a point counterclockwise about the origin, you can use the formulas x' = -y and y' = x.

To rotate a point counterclockwise about the origin, you can use the following formulas:
x' = x cos(θ) - y sin(θ)
y' = x sin(θ) + y cos(θ)

Here, (x', y') are the coordinates of the new point, (x, y) are the coordinates of the original point, and θ is the angle of rotation. For a counterclockwise rotation of 270 degrees, θ = 270°. However, it's important to note that a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. So, the formulas become:
x' = -y
y' = x

Therefore, the coordinates of the new point after rotating the original point (x, y) counterclockwise about the origin by 270 degrees are (-y, x).