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I think it is easier to find the roots of the characteristic polynomial without expanding and using the quadratic formula. They are sitting right in front of you.

You have to have both cos and sin to know the correct sense of the rotation.

EDIT: This is a real-world issue e.g. in Doppler radar. You need to duplicate the analogue circuitry, known as In-phase and Quadrature channels. In this case you need the quadrature channel to distinguish approaching and receding targets.

Actually for some purposes transcendental functions are overkill. You can parametrise

cos(theta) = (1-t^2)/(1+t^2)

sin(theta) = 2 t/ (1+t^2)

where

t = tan(theta/2)

t real on (-inf, inf)

There's a problem with theta = +/- 180 deg where t is singular. In practice just take cos(180 deg)= -1, sin(180 deg)=0.

EDIT: Or take homogenous parameters t0, t1 such that t=t1/t0. Then

cos(theta) = (t0^2-t1^2)/(t0^2+t1^2)

sin(theta) = 2 t0 t1/(t0^2+t1^2)

The 180 deg case is (t0, t1) = (0, 1)

For proving that R_theta is a rotation matrix, isn’t it easier to write v = (r cos(phi), r sin(phi)) and apply trigonometric identities to show that R_theta v = (r cos(phi+theta), r sin(phi+theta))? That gets you around the issues with whether cos is properly injective too.

Nice video Michael! Another nice picture for this is the Bloch sphere / Pauli algebra one from quantum mechanics, though that might be a stretch for Linear Algebra students. There the standard rotation matrix shows up as an exponential of the Pauli Y matrix (rotation around y axis of the Bloch sphere), and the two orthogonal eigenvectors reappear as the two antipodal points on the y axis of the sphere.

Please just keep doing your favourite results

At 8:30, since cos() is even, the rotation might be cw or ccw by theta.

Love the video! Keep it up!

Neat. But there's obvious generalization to higher dimension, afaic. Save for perhaps 3, and 7.

Plane rotations can be viewed as a 1D projective space, i.e. line, known as an S1, with basis points 1 and k where k is a quaternion. The general rotation is

u = cos(theta/2) + k sin(theta/2)

To compound rotations u and v you take the product vu. v here can be understood as a projectivity on the S1.

To rotate a plane point (x,y) by u you make the quaternion

q = 1 + ix + jy

and then

q -> u q u*

where * is quaternion conjugation.

This generalises pretty well. In fact ALL lines in the projective 3-space, or S3, of the quaternions represent rotations about some axis in Euclidean 3-space (strictly an axis through the origin).

Sorry to unload on Michael's channel, I just find it a fascinating topic.

I'd love to see your take on 3D rotations, as represented by quaternions.

8:15 "theta is between zero to 2pi and on that region cos is injective"???

Reminds me of Ms Polly DeYoung in the 11th grade. But nothing new.

what is the "second math major channel"

22:02 8 → ∞