Tower 1 is located 100m directly west of Tower 2. From tower 1,
PERSON A looks (S 42 degrees E)and at an angle of depression of 50 degrees, can see PERSON B on the ground. PERSON B looks N 54 deg E and at an angle of elevation of 62 deg, can see PERSON C in Tower 2. AT what angle of depression, to the nearest degree, is PERSON C looking over at PERSON A?
Eventually, we will simply break this problem into three separate triangles, one with A and B, one with B and C, and one with C and A – all of which we will look at from a “side view” (also known as an elevation view).
First, however, we’ll look at a “bird’s eye view” of the situation, i.e., looking down from above (also known as a plan view). In the plan view, A, B, and C form a triangle with angle A = 48°, B = 96°, and C = 36°. Side AC = 100m. Be sure to draw a picture of this. From this information we can determine by using the Law of Sines that AB = 59.1m and BC = 74.72m. These are the horizontal distances.
Now let’s look at A and B from the side. Let X be a point directly beneath A and at the same elevation as B. The distance from X to B is the 100m determined above. Since the angle of depression from A to b is 50°, this is also the angle of elevation from B to A. Draw a picture of this triangle and label all this information on it. From this we can determine the length of XA, which is the height of A above B, to be
XA = 59.1*tan50° = 70.435m
Similarly, we can also construct a triangle with B and C and a point directly below C, which we will call Y. the distance from B to Y is the 74.72m determined above. The angle of elevation from B to C is 62°. From this we can determine the length of YC, which is the height of C above B, to be
YC = 74.72*tan62° = 140.535m
The difference in height between C and A is therefore
YC – XA = 140.535 – 70.435 = 70.1m
The horizontal difference between them is 100m. Draw this triangle also. The angle of elevation from A to B equals the angle of depression from B to A:
@ = arctan (70.1/100) = 35°
Please check my work. I hope this helps.
