What is the new point when you rotate a geometric figure with 5, 8270 degrees clockwise about the origin minus 8, - 5 - 5, 8 8, 5 8, - 5
To rotate a geometric figure, we need to find the new coordinates of each point after the rotation. The rotation is given as 5,8270 degrees clockwise about the origin.
Let's find the new coordinates for each point:
1. For the origin (0,0), the new coordinates will still be (0,0) since rotating around the origin does not change its position.
2. For the point (-8, -5), rotating it 5,8270 degrees clockwise means we need to find the new angle produced by this rotation. We can calculate the new angle by subtracting 5,8270 degrees from the original angle (-8, -5) forms with the positive x-axis.
Using trigonometry, we can find that the original angle is approximately 142.31 degrees. Subtracting 5,8270 degrees from it gives us a new angle of -5724.69 degrees (approximately).
Now we can calculate the coordinates of the new point using the new angle:
x = length * cos(angle) = √((-8)^2 + (-5)^2) * cos(-5724.69 degrees) ≈ 9.05 * (-1) ≈ -9.05
y = length * sin(angle) = √((-8)^2 + (-5)^2) * sin(-5724.69 degrees) ≈ 9.05 * (-0.04) ≈ -0.36
So, the new coordinates for (-8, -5) after the rotation are approximately (-9.05, -0.36).
3. For the point (8, 8), the same process as above applies. The original angle formed with the positive x-axis is approximately 45 degrees. Subtracting 5,8270 degrees from it gives us a new angle of -5,824.69 degrees (approximately).
Calculating the new coordinates:
x = length * cos(angle) = √(8^2 + 8^2) * cos(-5824.69 degrees) ≈ 11.31 * (0.03) ≈ 0.34
y = length * sin(angle) = √(8^2 + 8^2) * sin(-5824.69 degrees) ≈ 11.31 * (-1) ≈ -11.31
So, the new coordinates for (8, 8) after the rotation are approximately (0.34, -11.31).
4. For the point (8, -5), following the same steps as above:
Original angle with the positive x-axis is approximately -30.96 degrees. Subtracting 5,8270 degrees gives a new angle of -5898.96 degrees (approximately).
Calculating the new coordinates:
x = length * cos(angle) = √(8^2 + (-5)^2) * cos(-5898.96 degrees) ≈ 9.43 * (-0.33) ≈ -3.11
y = length * sin(angle) = √(8^2 + (-5)^2) * sin(-5898.96 degrees) ≈ 9.43 * (0.018) ≈ 0.17
So, the new coordinates for (8, -5) after the rotation are approximately (-3.11, 0.17).
To summarize, the new coordinates after rotating the given points 5,8270 degrees clockwise about the origin are as follows:
Origin (0, 0): (0, 0)
(-8, -5): approximately (-9.05, -0.36)
(8, 8): approximately (0.34, -11.31)
(8, -5): approximately (-3.11, 0.17)