Norm with supremum

Question

hope your having a good day, I have stumbled across this question in Topology, which is to show that:

N((x, y)) = Sup|xcost+ysint|

For all t belong in [0,2π] and (x, y) in R^2 is a norm.

Hooe you could aid me in this and thank you

Answer 1

HINT: Show that $N(\langle x,y\rangle)=\sqrt{x^2+y^2}$, the usual Euclidean norm. You may find it useful to realize that if $\mathbf{e}_t=\langle\cos t,\sin t\rangle$ is the unit vector in the direction $\theta=t$, then $x\cos t+y\sin t$ is the dot product $\langle x,y\rangle\cdot\mathbf{e}_t$; think about the geometric interpretation of that dot product.