Suppose there is another x'y'-coordinate system that has the same origin as the xy-coordinate system, and θ is the angle from the positive x-axis to the positive x'-axis. If there is a point (x, y) in the xy-coordinate system, and a point (x', y') in the rotated x'y'- coordinate system, then
x = x' cos θ - y' sin θ and
y = x' sin θ + y' cos θ (rotation of axis formulas)
Since the rotation of 60 degrees clockwise, is the same as the rotation of 300 degrees anticlockwise,
then cos 300ᵒ = cos (-60ᵒ) = 1/2 and sin 300ᵒ = sin (-60ᵒ) = -√3/2 (only cosine is positive in the IV quadrant).
So we need to express x' and y' in terms of x and y.
x = x' cos θ - y' sin θ
x = (1/2)x' - (-√3/2)y' multiply by 2 each term to both sides
2x = x' + (√3)y' subtract (√3)y' to both sides
x' = 2x - (√3)y'
y = x' sin θ + y' cos θ
y = (-√3/2)x' + (1/2)y' multiply by 2 to both sides
2y = (-√3)x' + y' add (√3)x' to both sides
y' = (√3)x' + 2y
so that,
x' = 2x - (√3)y' replace y' by (√3)x' + 2y
x' = 2x - √3[(√3)x' + 2y]
x' = 2x - 3x' - 2√3y add 3x' to both sides
4x' = 2x - 2√3y divide by 4 to both sides
x' = (1/2)x - (√3/2)y
and
y' = (√3)x' + 2y replace x' by (1/2)x - (√3/2)y
y' = (√3)[(1/2)x - (√3/2)y] + 2y
y' = (√3/2)x - (3/2)y + 2y
y' = (√3/2)x + (1/2)y
Thus, the rotated point (if the angle of rotation about the origin is 60 degrees clockwise) is [(1/2)x - (√3/2)y, (√3/2)x + (1/2)y].
You dont, its just 90 degrees 3 times..
Move it 3 times* * * * *or once in the anti-clockwise direction.
To rotate a figure 180 degrees clockwise about the origin you need to take all of the coordinates of the figure and change the sign of the x-coordinates to the opposite sign(positive to negative or negative to positive). You then do the same with the y-coordinates and plot the resulting coordinates to get your rotated figure.
I dont really know if this is right but i think to do this problem you have to take a point then rotate the paper counter clockwise around the origin then you have a new point which is called a prime. Then reflect it over the y axis on the graph.
rotate it 90 degrees
You dont, its just 90 degrees 3 times..
Move it 3 times* * * * *or once in the anti-clockwise direction.
To rotate a figure 180 degrees clockwise about the origin you need to take all of the coordinates of the figure and change the sign of the x-coordinates to the opposite sign(positive to negative or negative to positive). You then do the same with the y-coordinates and plot the resulting coordinates to get your rotated figure.
No, only their positions will change.
To rotate a figure 90 degrees clockwise about the origin, simply swap the x and y coordinates of each point and then negate the new y-coordinate. This is equivalent to reflecting the figure over the line y = x and then over the y-axis.
To rotate a figure 180 degrees clockwise, you can achieve this by first reflecting the figure over the y-axis and then reflecting it over the x-axis. This double reflection effectively rotates the figure 180 degrees clockwise around the origin.
multiply the coordinates by -1.
360 degree rotation (clockwise or anticlockwise) leaves any figure in exactly the same position as it was at the start. So YOU DO NOTHING.
I dont really know if this is right but i think to do this problem you have to take a point then rotate the paper counter clockwise around the origin then you have a new point which is called a prime. Then reflect it over the y axis on the graph.
rotate it 90 degrees
Visualize a capital "N." Rotated 90 degrees counter-clockwise (a quarter turn to the left) it would look like a capital "Z."
The x,y origin is 0,0