Question

A Texas rancher wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure below, where A = 4.90 km and θC = 15°.

Answer 1

The answer:
the full question is as follow:
 A Texas rancher wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure below , where A = 4.90 km and θC = 15°. He then correctly calculates the length and orientation of the fourth side D. What is the magnitude and direction of vector D? 

As shown in the figure, 
A + B + C + D = 0, so to find the magnitude and direction of vector D, we should follow the following method:
 D = 0 - (A + B + C) , 
let  W = - (A + B + C), so the magnitude and direction of vector D is the same of the vector W characteristics

Magnitude
 A + B + C =  (4.90cos7.5 - 2.48sin16 - 3.02cos15)I
+ (-4.9sin7.5 + 2.48cos16 + 3.02sin15)J
= 1.25I +2.53J
the magnitude of W= abs value of (A + B + C) = sqrt (1.25² + 2.53²)
= 2.82
 
the direction of D can be found by using Dx and Dy value
we know that     tanθo = Dx / Dy = 1.25 / 2.53 =0.49
tanθo =0.49 it implies θo = arctan 0.49 = 26.02°

direction is 26.02°