# How to find the angle and direction of a rotation matrix in 2D?

• Oct 17th 2012, 11:30 AM
LoneWolf
How to find the angle and direction of a rotation matrix in 2D?
If I have a rotation matrix:
(-0.7 0.7)
(-0.7 -0.7)

how can I get the angle and the direction of it?

If I do:
(cos-1(-0.7) -sin-1(0.7)
(sin-1(-0.7) cos-1(-0.7)

I get:
(134 -44)
(-44 134)

which are two different angles, 135 and -44 in degrees, how do I know which one is the answer and what direction the rotation is happening?

I couldn't find anything on the internet, so any links explaining how to get the angle and the direction would be grateful.
• Oct 17th 2012, 12:20 PM
earboth
Re: How to find the angle and direction of a rotation matrix in 2D?
Quote:

Originally Posted by LoneWolf
If I have a rotation matrix:
(-0.7 0.7)
(-0.7 -0.7)

how can I get the angle and the direction of it?

If I do:
(cos-1(-0.7) -sin-1(0.7) <--- typo!
(sin-1(-0.7) cos-1(-0.7)

I get:
(134 -44)
(-44 134)

which are two different angles, 135 and -44 in degrees, how do I know which one is the answer and what direction the rotation is happening?

I couldn't find anything on the internet, so any links explaining how to get the angle and the direction would be grateful.

1. I assume that your matrix looks like this:

$\left(\begin{array}{cc} -\frac12 \sqrt{2}&\frac12 \sqrt{2} \\ -\frac12 \sqrt{2} & -\frac12 \sqrt{2} \end{array}\right)$

2. You should know that $\sin(135^\circ) = \sin(45^\circ)$

You now have all informations to determine the angle of rotation.