Question

One day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From your starting point, you travel 2.00 miles down the 1st trail. Then you turn to your left by 30.0° to follow a 2nd trail for 1.20 miles. Next, you turn to your right by 160° and follow a 3rd trail for 1.70 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point?

Answer 1

Let the starting point be the origin and the first trail be due east (positive x axis)
Turning left 30 deg puts you at 30 deg above east which is 30 degs
Turning right 160: right 30 puts you back at 0 deg = 360 deg leaving 130 more clockwise which means 360-130=230 deg. This would be the same direction as turning left 360-160 = 200 and 200+30=230 so it checks.
Sum the horizontal components
2cos0 +1.2cos30+1.7cos230 = 1.946 mi east of the starting point
Sum the vertical components
2sin0+1.2sin30+1.7sin230 = -0.7022 south of starting point
Find distance from origin:
Distance = sqrt[1.946*1.946+.7022*.7022] = 2.0688 miles
at arctan(-.7022/1.946) = -19.8416 deg
Turn 180 degrees to you right and walk back 2.0688 miles.