Question

a hunter hidden in a tree 90 feet abovethe ground spots two deer in the forest one is due west at an angle of depresion of 34 and the other is due east at an angle of depression of 58 how far apart are the two deer

Answer 1

You know the height of the triangle, 90 ft, and the depression angle from the top gives you the same angle on the ground. So calculate it as two separate triangles and add the base amounts together.

First triangle to the west:
90 ft high, base angle is 34 degrees. You want to know the horizontal distance on the ground and you know the vertical height and can care less about the hypotenuse. So you choose the tangent of 34 degrees to find out the horizontal line length because tan = opposite angle / adjacent angle. So draw a triangle with the opposite side equal to 90 ft

tan34 = 0.6745
tan 34 = opposite / adjacent
0.6745 = 90ft / adj
adjacent = 90 ft / 0.6745 = 133.4 ft, horizontal distance from hunter to west deer

Now do west triangle same way:

tan 58 = 1.6
tan 58 = opposite / adjacent
1.6 = 90ft / adj
adjacent side = 90ft / 1.6 = 56.25 ft , horizontal distance from hunter to east deer

add two distances together to get total distance between two deers:

56.25 ft + 133.4 ft = 189.65 ft