How to derive the sine formula from the cross product

Question

Use the cross product to show that sinthetaA÷vector BC = Sin thetaB÷vector AC

Answer 1

Equate the area of triangle

$$\lvert\vec{AC}\times\vec{AB}\rvert=\lvert\vec{BA}\times\vec{BC}\rvert\\ \lvert\vec{AC}\rvert\ \lvert\vec{AB}\rvert \sin{\theta_{A}}=\lvert\vec{BA}\rvert\ \lvert\vec{BC}\rvert \sin{\theta_{B}}\\ \lvert\vec{AC}\rvert\ \sin{\theta_{A}}= \lvert\vec{BC}\rvert \sin{\theta_{B}}$$