trigonometry: Quick Facts

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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.

Diagram explaining the Egyptian seked, showing a pyramid with labeled dimensions and a triangle illustrating the run and rise. A formula box defines seked as the ratio of run to rise, with an example calculation.

A triangle is inscribed in a circle, with vertices labeled A, B, and C. The central angle theta is shown at the circle's center, formed by two radii meeting at points A and B, illustrating the relationship between the angle and the chord AB.

Diagram of a circle with center O showing two radii labeled r, a central angle A, and a chord labeled c opposite the angle, illustrating the trigonometric relationship between the chord length and the angle.

Diagram illustrating general angles in trigonometry, with two rays labeled QV and PV forming an angle at point V. A red arrow shows a positive angle from Q to P, and a dashed arc shows a negative angle in the opposite direction.

Two geometric diagrams show how to add and subtract angles. On the left, angle A and angle B are added at point V, forming angle A+B between lines VP and VR. On the right, angle B is subtracted from angle A, forming angle A-B between lines VP and VQ, with arrows indicating the direction of angle measurement.

A diagram shows an angle in standard position on a coordinate plane, with its initial side along the x-axis and terminal side passing through point Q at coordinates (x, y). The angle A is marked between the x-axis and the terminal side, with line segments labeled x, y, and r forming a right triangle.

A labeled triangle with vertices A, B, and C, sides a, b, and c, and a red line segment h drawn from vertex C perpendicular to side c, representing the height.

Four graphs display the curves of sine, cosine, tangent, and cotangent functions, each labeled with x- and y-axes, key angles in radians, and typical wave or asymptote patterns.

A diagram shows a point P in a plane with its position defined by both Cartesian coordinates (x, y) and polar coordinates (r, theta), forming a right triangle with the x-axis and y-axis labeled.

How did Ptolemy explain retrograde motion?

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Topics

analytic trigonometry spherical trigonometry plane trigonometry spherical triangle triangle trigonometric function trigonometry table mathematics analysis probability theory cryptology computer science combinatorics algebra statistics geometry automata theory set theory

Key People

Engraving of six astronomers, with Hipparchus seated at center and, from left to right, Galileo, Johannes Hevelius, Tycho Brahe, Nicolaus Copernicus, and Ptolemy standing, each holding scientific instruments.

Hipparchus

Greek astronomer

Leonhard Euler

Swiss mathematician

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Regiomontanus

German mathematician

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Abū al-Wafāʾ

Persian mathematician

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François Viète

French mathematician

James Gregory.

James Gregory

Scottish mathematician and astronomer

Equations written on blackboard

Numbers and Mathematics