solving oblique triangles?

Question

Kurl wans to sail his boat from a marina to an island 15 miles east of the marina. Aling the course, there are several small islands they must avoid. He sails first on a heading of 70 degrees and then on a heading of 120 degrees(remember the headings angle measures rotated clockwise from the north). What is the total distance he travels before reaching the island?

Answer 1

In the triangle ABC, A is the marina and C the island. B is the point of diversion. Angle BAC=90-70=20 degrees. At point B the heading changes from 70 to 120, a difference of 50 degrees. So 50 degrees is the external angle at B. The interior angle ABC=180-50=130, so angle BCA is 180-(130+20)=30. AC=15 miles. We can use the sine rule to find the other two sides and therefore we can calculate the distance. AB/sin30=15/sin130=BC/sin20. AB=15sin30/sin130=9.79m; BC=15sin20/sin130=6.70m. Total distance=AB+BC=15(sin20+sin30)/sin130=16.49 miles.