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.
rotate it 90 degrees
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.
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.
Take any one point on the figure. Draw a line from it to the origin. At the origin measure an angle of 90 degrees (right angle) in a clockwise direction. Draw a line from the origin at this new angle and of the same length as the original angle. Repeat this process for the other points in the figure. NB Be careful, there will be numerous lines from the origin. At the end points of the new lines, connect up to reveal the origin figure ,but rotated 90 degrees - clockwise.
multiply the coordinates by -1.
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
360 degree rotation (clockwise or anticlockwise) leaves any figure in exactly the same position as it was at the start. So YOU DO NOTHING.
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
The same as 180 degrees clockwise. What do you mean "the answer to"?