Find the coefficients $A_n$ if $T(x, 0) = \sin x \cos x$

Question

Given that $T(x,t) = \sum_{n=1}^\infty A_n\sin(nx)e^{-n^2t},$ find the coefficient $A_n$ given that $T(0,t)=0=T(\pi,t)$ and that $T(x,0)$.

Any help would be great

Answer 1

Use the trigonometric identity $$\sin x \cos x = \dfrac 1 2 \sin(2x)$$ (the double-angle formula for the sine function).

That means only the term $n=2$ is not zero. So $A_2 = \text{ ?}$