WPP #13 and 14: Unit P Concepts 6 & 7: Applications with Law of Sines & Law of Cosines

This WPP 13 & 14 was made in collaboration with Leo Escutia. Please visit the other awesome post on their blog by going here.
  

     One day Robin decided to go fishing but out at sea. He knew that his long time friend was fisher too and he had a boat. So his friend Gabriel took his to a good fishing spot off the coast. Hours passed but when they decided to head back, the engine failed. They radioed for help and two Coast Guard station responded back. They needed their exact location but the boat was acting weird and couldn't give the exact distance. Station B is due south of Station A and they are 100 miles apart. From what they can tell, the boat is S 45 degrees W of Station A, and N 65 degrees W of Station B. How far is each station to the boat?

Here we have it drawn to show how the triangle is suppose to look like. This a ASA problem since we give two angles and one side in between them.  Since we give two angles, the third missing side will be70 degrees.

We used the Law of Sine to solve for side "a". Angle C, which is where the boat is, will be our bridge to solve for side "a' and side "b". Sin 70/ 100 = Sin 45/ a, we cross multiply to give us a(sin 70) = 100(sin 450). We divide sin 70 to both sides to cancel sin 70 on one side so that it will be a = 100(sin 45)/ sin 70. Station A is 75.2 miles away.

Like I said, sin 70/ 100 is our bridge and it will be used to fine side "b". So sin 70/100 = sin 65/b, we cross multiply to give us b(sin 70) = 100(sin 65). We divide sin 70 to both sides so that sin 70 cancels on one side so that we can have b = 100(sin 65)/ sin 70. Station B is 96.4 miles away from the boat.

     Once they got rescued and taken back to Station A they are taken back home. They both leave the station at the same time, they diverge an angle of 100 degrees. If Robin is 4 miles away from the station and Gabriel is 5.5 miles away too, then how far away are they from each other?

With it's picture and work done together, we will use the Law of Cosine to find the missing side. It will be a^2 = 4^2 + 5.5^2 - 2(4)(5.5) cos 100. We just plug that in to our calculator and they are 7.3 miles apart.