Trigonometry (11th Edition) Clone

The bearing the pilot should fly is $306^{\circ}$ The ground speed will be 523.8 mph
Let $a = 520~mph$ Let $b = 37~mph$ Let $c$ be the resultant of the two vectors $a$ and $b$. Note that the magnitude of $c$ is the ground speed. Let angle $A$ be the angle that subtends side $a$. Then $A = 32^{\circ}+50^{\circ} = 82^{\circ}$ We can use the law of sines to find the angle B between the vectors a and c: $\frac{a}{sin~A} = \frac{b}{sin~B}$ $sin~B = \frac{b~sin~A}{a}$ $B = arcsin(\frac{b~sin~A}{a})$ $B = arcsin(\frac{(37)~sin~82^{\circ}}{520})$ $B = arcsin(0.07046)$ $B = 4.0^{\circ}$ The bearing the pilot should fly is $310^{\circ}-4.0^{\circ} = 306^{\circ}$ We can find the angle $C$: $A+B+C = 180^{\circ}$ $C = 180^{\circ}-A-B$ $C = 180^{\circ}-82^{\circ}-4.0^{\circ}$ $C = 94^{\circ}$ We can use the law of sines to find $c$: $\frac{c}{sin~C} = \frac{a}{sin~A}$ $c = \frac{a~sin~C}{sin~A}$ $c = \frac{(520~mph)~sin~94^{\circ}}{sin~82^{\circ}}$ $c = 523.8~mph$ The ground speed will be 523.8 mph