Write down the matrix corresponding to a rotation over an angle phi in the plane

Question

Write down the matrix corresponding to a rotation over an angle phi  in the plane R2. Now write down the matrix corresponding to the linear transformation which is the composition of a rotation in the plane over an angle phi followed by a rotation in the plane over an angle theta . To which linear transformation does this matrix correspond? Deduce from this formulas for cos(theta + phi ) and sin( theta+phi ) in terms of the cosines and sines of  theta and phi . Check that the order in which we performed the rotations does not matter. Aside For 3-dimensional rotations the order does matter, think of the Rubik's cube for instance

Explanation / Answer

Matrix fro rotation of phi is given by

cos(phi) -sin(phi)

sin(phi) cos(phi)

Composition of rotation for phi and rotation for theta is

cos(theta) -sin(theta)

sin(theta) cos(theta)

multiplied by

cos(phi) -sin(phi)

sin(phi) cos(phi)

product matrix is

cos(theta)cos(phi)-sin(theta)sin(phi) -cos(theta)sin(phi)-sin(theta)cos(phi)

sin(theta)cos(phi)+cos(theta)sin(phi) cos(theta)cos(phi)+sin(theta)sin(phi)

this matrix actually correspods to rotation by (theta+phi) and hence comparing the elements of matrix we get

cos(theta+phi)=cos(theta)cos(phi)-sin(theta)sin(phi)

and

sin(theta+phi)=sin(theta)cos(phi)+cos(theta)sin(phi).

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