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How does triangulation use a known baseline and measured angles to map land?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

Triangulation is a fundamental technique in surveying used to map land by establishing a framework of triangles across a geographic area.12

The process relies on the principles of plane trigonometry, which dictate that if one side and two angles of a triangle are known, the remaining sides and angles can be calculated. The method functions as follows:13

Establishing the Baseline: A surveyor selects a base line of known length, which serves as the starting side for the first triangle.14
Measuring Angles: Using a theodolite, the surveyor measures the two angles at the endpoints of the baseline that point toward a remote, third location.14
Calculating Distances: With one side and two angles known, the lengths of the other two sides are determined through trigonometric calculations.12
Expanding the Network: This process is repeated by using the newly calculated sides as baselines for adjacent triangles. By constructing a continuous chain or network of these triangles, surveyors can map large areas, ensuring that different sections fit together accurately without gaps or overlaps.142