Etymology
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trigonometry (n.)

"branch of mathematics that deals with relations between sides and angles of triangles," 1610s, from Modern Latin trigonometria (Barthelemi Pitiscus, 1595), from Greek trigonon "triangle" (from tri- "three" (see tri-) + gōnia "angle, corner" (from PIE root *genu- (1) "knee; angle") + metron "a measure" (from PIE root *me- (2) "to measure").

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trigonometric (adj.)
1811; see trigonometry + -ic. Related: Trigonometrical (1660s).
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*genu- (1)

Proto-Indo-European root meaning "knee; angle."

It forms all or part of: agonic; decagon; diagonal; geniculate; genuflect; genuflection; -gon; goniometer; heptagon; hexagon; knee; kneel; octagon; orthogonal; pentagon; polygon; trigonometry.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit janu, Avestan znum, Hittite genu "knee;" Greek gony "knee," gōnia "corner, angle;" Latin genu "knee;" Old English cneo, cneow "knee."

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*me- (2)
*mē-, Proto-Indo-European root meaning "to measure." Some words may belong instead to root *med- "to take appropriate measures."

It forms all or part of: amenorrhea; centimeter; commensurate; diameter; dimension; gematria; geometry; immense; isometric; meal (n.1) "food, time for eating;" measure; menarche; meniscus; menopause; menses; menstrual; menstruate; mensural; meter (n.1) "poetic measure;" meter (n.2) unit of length; meter (n.3) "device for measuring;" -meter; Metis; metric; metrical; metronome; -metry; Monday; month; moon; parameter; pentameter; perimeter; piecemeal; semester; symmetry; thermometer; trigonometry; trimester.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit mati "measures," matra "measure;" Avestan, Old Persian ma- "to measure;" Greek metron "measure," metra "lot, portion;" Latin metri "to measure."
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cosecant (n.)

in trigonometry, 1706, from co, short for complement, + secant.

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cotangent (n.)

in trigonometry, "the tangent of the complement of a given angle," a contraction of co. tangent, abbreviation of complement + tangent (n.).

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tangent (n.)

1590, one of the three fundamental functions of trigonometry, from tangent (adj.). From 1650s as "a tangent line." Figurative use of on a tangent is from 1771.

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cosine (n.)

one of the three fundamental functions of trigonometry,  1630s, contraction of co. sinus, abbreviation of Medieval Latin complementi sinus (see complement + sine). The word was used in Latin c. 1620 by English mathematician Edmund Gunter.

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secant (n.)

one of the fundamental functions of trigonometry, 1590s, from Latin secantem (nominative secans) "a cutting," present participle of secare "to cut" (from PIE root *sek- "to cut"). First used by Danish mathematician Thomas Fincke in "Geometria Rotundi" (1583). Related: Secancy ("state of being a secant").

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