**A BIT OF GEOMETRY by Fredric M. Menger, Emory University, Atlanta, GA, USA**

You are standing on the ocean shoreline at point-

**P**while looking at a ship**S**far out in the water. You want to know**x**, your distance from the ship. So you walk along the beach until you come to point-**A**and measure the angle-**α**. Then you walk back along the shoreline to point-**B**and obtain a reading of angle-**β**. Assume you have also measured the**A/B**distance referred to as**d**. How do you determine**x**, the distance from**P**to the ship**S**(assuming you have a calculator with trigonometric functions with you)?
Clearly,

You know that (by definition) tan

Substituting:

and:

Since you have measured

**d = d1 + d2**You know that (by definition) tan

**α = x/d1**and tan**β = x/d2**(where “tan” refers to the tangent of the angle). Rearranging,**d1 = x/**tan**α**and**d2 = x/**tan**β**.Substituting:

**d = x/**tan**α**+**x/**tan**β**or**d = x**(1/tan**α**+ 1/tan**β**)and:

**x = d/**(1/tan**α**+ 1/tan**β**)Since you have measured

**d**, and your calculator gives you the tangents of the angles, you can determine**x**, the distance across the water from point**P**to the ship**S**.
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