Why the tangent bundle of a smooth manifold is an oriented manifold?

Question

I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you!

For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.

Answer 1

Hint: Take an atlas on $M$, say $\{(U,\varphi)\}$ and look at the corresponding atlas on $TM$, $\displaystyle \left(U,\frac{\partial}{\partial \varphi}\right)$. What is the determinant of these overlap maps' Jacobians?