## Conceptual Physics (12th Edition)

In the same time that it takes sound to travel from point A to point C, the plane has moved from point A to point B. Triangle ACB is a special 45-45-90 right triangle. The distance traveled by the plane, AB, is $\sqrt{2} \times AC$. An internet search for the term "Mach angle" shows that generally, the sine of the half-angle gives the ratio of the speed of sound to the speed of the plane. $$sin(45^{\circ}) = \frac{1}{\sqrt{2} } = \frac{v_{sound}}{v_{plane}}$$ This gives the same answer.