For example, let us consider our old friend $$\vec{A} = 5 \hat{i} + 7 \hat{j}$$. Clearly, $$A_x = 5, A_y = 7$$. The magnitude is thus
$$\sqrt{5^2 + 7^2} = 8.6$$
$$\tan^{-1} \left(\frac{A_y}{A_x}\right) = \tan^{-1} \left(\frac{7}{5}\right) = 54^{\circ}$$