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How does trigonometry help with navigation and mapmaking?

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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.

Trigonometry is essential to navigation and mapmaking because it allows for the precise calculation of distances and angles that cannot be measured directly.12

Navigation

In navigation, trigonometry is used to determine the position of ships and aircraft. By applying the laws of plane trigonometry, navigators use a technique called triangulation, where measuring one side (a baseline) and two angles of a triangle allows for the calculation of the remaining sides and angles. For longer distances on Earth's surface, spherical trigonometry is employed. This involves spherical triangles formed by great circle arcs, utilizing formulas such as the spherical law of cosines to calculate accurate travel distances.234

Mapmaking

Mapmaking, or cartography, relies on these same principles to represent the Earth's features accurately. By establishing a network of connected triangles across a landscape, surveyors can map large areas with high precision. Historically, this mathematical foundation enabled early astronomers and geographers to refine global measurements and create more reliable representations of the world.1325