A man standing on the top of a hill and sees a flagpole he knows is 45 feet high. The angle of depression to the bottom of the pole is 12 degrees, and the angle of elevation to the top of the pole is 16 degrees. Find his distance from the pole

160.44 feet

Explanation:

check the attachment for the diagram of the set up for proper clarification.

From the diagram, the man's distance from the pole is labelled AB.

To get AB =, we need to get side CB first which is equal to ED i.e ED = CB.

From ΔCDE, opposite side = 45 feet and the adjacent is ED.

USing TOA, according to trig function SOH, CAH, TOA;

tan 16° = opp/adj = BD/ED

tan 16° = 45/ED

ED = 45/tan 16°

ED = 156.93 feet

Since ED = CB, hence CB = 156.93 feet

From ΔABC, adj = CB and hypotenuse = AB. According to CAH;

cos 12° = adj/hyp = CB/AB

cos 12° =  156.93/AB

AB =  156.93/cos 12°

AB = 160.44 feet

Hence the man's distance from the pole is 160.44 feet

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