Let $\{W_t\}_{t\geq0}$ be a Brownian motion. Define $Y_t=e^{\frac{t}{2}}\sin(W_t)$. Show that $\{Y_t\}_{t\geq0}$ is a martingale

Question

I know one way to do this but my TA mentioned that there is a method that involves applying Ito's formula. I did this and obtained: $$dY_t=e^{\frac{t}{2}}\cdot \cos(W_t)\:dW_t$$ I know that $W_t$ is a martingale by nature of the fact that it is a Brownian motion. But how is it possible to conclude that $Y_t$ is a martingale as well?

I am very new to It's formula. If you can provide me with an understanding with why we decide to apply it in the first place I would be very appreciative. In the world of normal calculus I would never think to take a derivative here so why do it when dealing with stochastic calculus? I know this shows how little I understand the material so any explanation will have me in a better position.

Answer 1

$Y_t=\int \text{something } dW_t$ and is thus a martingale. See: https://fabricebaudoin.wordpress.com/2012/08/30/lecture-17-some-properties-of-the-ito-integral/ for example.