Two scouts are in contact with home base. Scout A is 15 km from home base in a direction 30 degrees north of east. Scout B is 12 km from home base in a direction 40 degrees west of north. How far is scout B from scout A?
Scout A, home base and a point directly south A form a right triangle with a 15 km hypotenuse and an interior angle of 30 degrees. Scout A is `15 sin 30` km north, or 7.5 km. The scout is `15 cos 30` or 13km East.
Scout B, home base and a point directly north from home base form another right triangle with a hypotenuse of 12 km and an interior angle of 40 degrees. The scout is `12 cos 40` or 9 km North, and `12 sin 40` West.
From this information we can calculate the distance along North-South line between the two scouts is 9-7.5=1.5 km, and we can calculate the distance along East-West line between the two scouts is 13+8=21km. These lines are at right angles, so the straight line between A and B is the hypotenuse of a right Triangle with legs of 1.5 km and 21km. The Pythagorean theorem says that this hypotenuse will be `sqrt(21^2+1.5^2)=sqrt(441+2.25)~~21.05`
Scouts A and B are approximately 21.05 km apart.