Let $\lambda:[a,b]\to\mathbb{R^2}-\{0\}$ be a $C^1$ sectionally closed path and $\alpha\in\mathbb{R}$ a constant. Define $\mu:[a,b]\to\mathbb{R^2}-\{0\}$ by doing $\mu(t) = e^{i \alpha}\cdot \lambda(t) = (x(t)\cos \alpha - y(t)\sin \alpha, x(t)\sin \alpha+y(t)\cos \alpha)$, where $\lambda(t) = (x(t), y(t))$. Prove that the paths $\lambda$ and $u$ give the same number of turns around the origin.
I noted that $|\mu| = |e^{i\alpha}||\lambda(t)| = |\lambda(t)|$. I think it has to do with integration. I also had this exercise: Turns of closed path $\lambda$ around origin is related to area of paralelogram $\lambda(t), \lambda'(t)$ but this question comes before this exercise, so I must not use the formula present there.
However, I think it has to do with integration, because I'm studying Stieltjes integration, homotopic curves and so. How can I connect these results to solve this exercise?