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Trigonometric functions as segments

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Until the 20th century, trigonometric functions were conceived as the lengths of certain line segments associated with a point moving on a circle. The names of the functions describe the geometry.

Start with a point P on the circle, the radial line, through P and the center of the circle, and the perpendiculars through P to the x and y axes.

  • The tangent of P is the segment of the tangent line to the circle at (1,0) cut off by the radial line. ("Tangent" is Latin for "touching.")
  • The secant of P is the segment of the radial line cut off by the tangent line. ("Secant" is Latin for "cutting.")
  • The sine of P is the half the chord parallel to the tangent. ("Sine" is a Latin mistranslation of the Arabic word for "half chord.")
  • The "co-" in the cofunctions means the functions for the complementary angle to the angle between the x axis and the radial line. That is, to get the cofunctions, switch the roles of the axes and use the same constructions.

Susan Addington, Created with GeoGebra


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Last updated November 29, 2010

Copyright 2009-10 David Dennis and Susan Addington. All rights reserved.