Let $T(θ) : \Bbb R^2 → \Bbb R^2$ be the transformation that rotates each vector counterclockwise by angle $θ$.
(a) Write the standard matrix for $T(θ)$.
(b) Explain in words or pictures why $T(θ_1+θ_2) = T(θ_1) ◦ T(θ_2)$.
(c) Derive the angle sum formula for cosine and sine by finding the standard matrix for $T(θ_1+θ_2) = T(θ_1) ◦ T(θ_2)$; that is, prove that $\cos(θ_1+θ_2) = \cos(θ_1)\cos(θ_2)−\sin(θ_1) \sin(θ_2)$ and $\sin(θ_1+θ_2) = \sin(θ_1)\cos(θ_2) + \sin(θ_2)\cos(θ_1)$.
For part a - I'm not sure how to find the standard matrix when I don't know the initial values of the matrix before the vectors rotate.
Part b - I know $T(θ_1) ◦ T(θ_2)$ is the Hadamard product of $T(θ_1)$ and $T(θ_2)$, but I was never taught any properties to prove part b.
Part c - this problem completely confuses me.
Any help would be appreciated.
I'll say what you have to do.
a) Can you compute $T_\theta(1,0)$ and $T_\theta(0,1)$? You'll have to put their coordinates in the columns of the matrix. Make a drawing and see. For example, convince yourself that $T_\theta(1,0) = (\cos \theta, \sin \theta)$.
b) There's nothing to do with Hadamard product, it'll be ordinary matrix multiplication (maps composition). If you rotate by $\theta_1$, and then rotate by $\theta_2$, how much you rotated in the end?
c) Once you have $T_{\theta_1+\theta_2} = T_{\theta_1}\circ T_{\theta_2}$, compute the matrices of both sides and compare each entry. You'll get the formulas simple as that.