What is the good way to remember the signs of the rotational matrix?

Question

Recall rotational matrix in (x,y) is given by:

$R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$

For the life of me I cannot remember if the top right sin entry has a negative or positive sign, and I have known this matrix since 5 years ago (obviously I am a failure!!).

Sure one could always derive from first principle via:

But if someone just went up to you and asked you to write down the matrix, what is a good way in the least time write down the matrix and verify that it is correct (with the correct signs)?

Answer 1

Take $\theta=\pi/2$, and write down one of the options. It is fairly easy to check in this case whether you got a clockwise or anti-clockwise rotation.

Answer 2

Use complex numbers.

To rotate $x + iy$ through an angle $\theta$, multiply by $e^{i\theta} = \cos \theta + i \sin \theta$

$(x + iy)(\cos \theta + i \sin \theta) = (x \cos \theta - y \sin \theta) + i(x \sin \theta + y \cos \theta)$

which corresponds to

\begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{pmatrix}

Practically, all you really need to figure out is that the real part is $x \cos \theta - y \sin \theta$, or that the imaginary part is $x \sin \theta + y \cos \theta$.

I just remember that the signs in the top row are $1^2$ and $i^2$.

Answer 3

My idea is similar to Steven's.

Write complex numbers in vector forms on the complex plane. Rotating $\begin{pmatrix} \cos{\gamma}\\\sin{\gamma}\end{pmatrix}$ through angle $\theta$ counterclockwise, it is located at $\begin{pmatrix} \cos{(\gamma+\theta)}\\\sin{(\gamma+\theta)}\end{pmatrix}$. By the Trigonometric Addition Formulas, $$\cos{(\gamma+\theta)}=\cos{\gamma}\cos{\theta}-\sin{\gamma}\sin{\theta},$$ $$\sin{(\gamma+\theta)}=\sin{\theta}\cos{\gamma}+\sin{\gamma}\cos{\theta},$$ you can know the matrix $A$ which satisfies $$A \begin{pmatrix} \cos{\gamma}\\\sin{\gamma}\end{pmatrix} = \begin{pmatrix} \cos{(\gamma+\theta)}\\\sin{(\gamma+\theta)}\end{pmatrix}$$ is $$\begin{pmatrix} \cos{\theta}& -\sin{\theta} \\ \sin{\theta}& \cos{\theta} \end{pmatrix}.$$