x'y'-coordinate system

Question

The x'y'-coordinate system is rotated 𝜃 degrees from the xy-coordinate system. The coordinates of a point in the xy-coordinate system are given. Find coordinates of the point in the rotated coordinate system.

𝜃 = 60°, (4, 1)

Answer 1

To rotate a point counter-clockwise you can use the equations:

𝜃 = 60°, (4, 1)

x' = xcos(θ)-ysin(θ)

y' = xsin(θ)+ycos(θ)

x' = 4cos(60º)-sin(60º) = 4(1/2) - (√3/2) = (4-√3)/2 = 1.1339

y' = 4sin(60º)+cos(60º)=4(√3/2)+(1/2) = (4√3+1)/2 = 3.964

(1.1339, 3.964)

If you are familiar with matrices, you can use:

[x'] [cos(θ) -sin(θ)] [x]

[y'] =[sin(θ) cos(θ)][y]