In trigonometry, Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points , and two unknown points . From and an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of and . See figure; the angles measured are .
Since it involves observations of angles made at unknown points, the problem is an example of resection (as opposed to intersection).
Define the following angles:
As a first step we will solve for and . The sum of these two unknown angles is equal to the sum of and , yielding the equation
A second equation can be found more laboriously, as follows. The law of sines yields
Combining these, we get
Entirely analogous reasoning on the other side yields
Setting these two equal gives
Using a known trigonometric identity this ratio of sines can be expressed as the tangent of an angle difference:
Where
This is the second equation we need. Once we solve the two equations for the two unknowns , we can use either of the two expressions above for to find since is known. We can then find all the other segments using the law of sines.
We are given four angles and the distance . The calculation proceeds as follows:
Calculate or equivalently If one of these fractions has a denominator close to zero, use the other one.