It was mentioned in lecture that the columns of the matrix are how the basis vectors are transformed. Based on this explanation, I understand how we get the rotation matrix.

However, what is the trigonometric process behind going from the R_theta values of (cos(theta), sin(theta)) and (cos(theta + pi/2), sin(theta + pi/2)) to the matrix?

It was mentioned in lecture that the columns of the matrix are how the basis vectors are transformed. Based on this explanation, I understand how we get the rotation matrix.

However, what is the trigonometric process behind going from the R_theta values of (cos(theta), sin(theta)) and (cos(theta + pi/2), sin(theta + pi/2)) to the matrix?