Trig equations

Question

A boat sails on a bearing of 75° for 128miles and then turns and sails 203miles on a bearing of 191°. Find the distance of the boat from its starting point.

 

I am unsure how to solve this equation

Answer 1

It always helps to draw a picture.  Imagine this an an xy coordinate grid, where the boat starts at (0,0).  The boat sails 128 miles at an angle of 75°.  It's new (x,y) coordinates would be:

 

     (x₁,y₁) = (128·cos75°, 128·sin75°)

 

From (x₁,y₁), the boat then sails for 203 miles at 191°.  Its new and final position is:

 

      (x₂,y₂) = (x₁+203·cos191°, y₁+203·sin191°)

 

Use the distance formula to find the distance of (x₂,y₂) from the starting point (0,0).

 

     distance = ((x₂-0)²+(y₂-0)²)^1/2