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 about the origin, we can use the following formulas:
new_x = old_x * cos(angle) - old_y * sin(angle)
new_y = old_x * sin(angle) + old_y * cos(angle)
In this case, the angle of rotation is 270 degrees. Converting this to radians, we have:
angle = 270 * (π/180) = 3π/2
Now, substituting the given point (-4, 5) into the formulas, we can calculate the new point:
new_x = (-4) * cos(3π/2) - 5 * sin(3π/2)
= 4 * 0 - (-5) * 1
= 5
new_y = (-4) * sin(3π/2) + 5 * cos(3π/2)
= (-4) * (-1) + 5 * 0
= 4
Therefore, the new point after rotating (-4, 5) counterclockwise about the origin by 270 degrees is (5, 4).
To find the new point when the given point is rotated counterclockwise about the origin by 270 degrees, you can follow these steps:
Step 1: Determine the coordinates of the point with respect to the origin.
Since the given point is already in coordinates (-4, 5), its x-coordinate is -4, and its y-coordinate is 5.
Step 2: Apply the rotation formula.
To rotate a point counterclockwise about the origin, you can use the rotation formula:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this formula, (x, y) represents the original coordinates of the point, (x', y') represents the new coordinates after rotation, and θ represents the angle of rotation.
Step 3: Substitute the values into the rotation formula.
Let's substitute the values from the given point into the rotation formula with θ = 270 degrees (or π/2 radians):
x' = -4 * cos(π/2) - 5 * sin(π/2)
y' = -4 * sin(π/2) + 5 * cos(π/2)
Step 4: Simplify the expressions.
To evaluate the trigonometric functions of π/2, we have:
cos(π/2) = 0
sin(π/2) = 1
Substituting these values into the expressions:
x' = -4 * 0 - 5 * 1 = 0 - 5 = -5
y' = -4 * 1 + 5 * 0 = -4 + 0 = -4
Therefore, the new point after rotating the given point (-4, 5) counterclockwise about the origin by 270 degrees is (-5, -4).