System of trigonometric equations for angles A and B

Since $\sin(A + 2B) = \frac{\sqrt{3}}{2} \implies A + 2B = 60^\circ$ --- (1) Since $\cos(A + 4B) = 0 \implies A + 4B = 90^\circ$ --- (2) Subtracting (1) from (2): $2B = 30^\circ \implies B = 15^\circ$ Substitute $B = 15^\circ$ in (1): $A + 2(15^\circ) = 60^\circ \implies A = 30^\circ$ Thus, $A = 30^\circ$ and $B = 15^\circ$.