The point (3, 2) is rotated 30 degrees about the origin. What are the polar coordinates of this after rotation?

Question

Am I right in saying it's (root13, 30)? The angle rotated and the radius calculated as root(3^2 + 2^2)? I can convert between polar and cartesian, but I don't recall how to do rotation.

Answer 1

The rotation matrix about the origin is
R=
| cos(t) -sin(t) |
| sin(t) +cos(t) |
Any rotation through angle t of a point X(x,y) is then given by
RX
Put t=pi/6=30°
R=
| 0.8660 -0.5000 |
| 0.5000 +0.8660 |
and X(3,2)
Do the matrix multiplication:
RX=(1.598,3.232)

Note that a rotation about the origin does not change the radius, or distance from the origin, thus
2²+3²=13
1.598²+3.232²=13

Answer 2

you can also do the rotation simply by adjusting the angle. Rotate (√13,30) by 30, and you end up at (√13,60).

Now just convert that back to rectangular coordinates.