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In trigonometry and elementary geometry, triangulation is the process of finding coordinates and distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points, using the law of sines. (In the figure at right, the third angle of the triangle (call it θ) is known to be 180-α-β, or can be calculated as the difference between the two compass-bearings taken from A and B. The opposite-side for this angle is l, which is a known distance. Since, by the law of sines, the ratio sin(θ)/l is equal to that same ratio for the other two angles α and β, the lengths of any of the remaining two sides can be computed by algebra. Given either of these lengths, sine and cosine can be used to calculate the offsets in both the north/south and east/west axes from the corresponding observation point to the unknown point, thereby giving its final coordinates.) Some identities often used (valid only in flat or euclidean geometry): Triangulation is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision and gun direction of weapons. Many of these surveying problems involve the solution of large meshes of triangles, with hundreds or even thousands of observations. Complex triangulation problems involving real-world observations with errors require the solution of large systems of simultaneous equations to generate solutions. Famous uses of triangulation have included the retriangulation of Great Britain.
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