library

  Function and Relation Library

Trigonometric Functions:     Angle Definitions     Legs of A Triangle Definitions    
Sine     Cosine     Tangent     Secant     Cosecant     Cotangent

    Trig functions are functions of a number, y = f(x) and very often that number is an angle, y = f(x) or y = f( ).

Open this page & move point A
with the mouse to see all 6 functions move.

    As the point on the circle changes, the angle changes, since the point is on the terminal side of the angle.     As the angle changes, the length of each of the segments it determine changes.     As the length of the segment changes, the trig function changes.     Open the page in the box above to see this work!

    With a radius of 1:
  • The sine, is the length of AE.
  • The cosine is the length of BE.
  • The tangent is the length of FC.
  • The cosecant is the length of BG.
  • The secant is the length of BF.
  • The cotangent is the length of DG.

    For a radius other than one, the ratios below must be used to scale up or down the size of the circle and yield the value of the function.

    For more on this read The Unit Circle.


    Functions may also be defined in terms of x and y and r of right triangle ABE.

    The horizontal segment BE is leg x.

    The vertical segment AE is leg y.

    The segment AB is side r and the hypotenuse of the right triangle and is still the radius of the circle on which point (x,y) lies, and still the distance from the point to the origin.

    Defined in terms of x and y and r -- the lengths of BE, AE, and AB -- the sides of right triangle ABE, here are the trig functions.

  • The sine is the ratio AE/AB.
  • The cosine is the ratio BE/AB.
  • The tangent is the ratio AE/BE.
  • The cosecant is the ratio AB/AE.
  • The secant is the ratio AB/BE.
  • The cotangent is the ratio BE/AE.

    For info on each function, read below.


    The sine is the ratio of the y to r, the ratio of the vertical component to the radius.

    The sine is useful for describing natural phenomena and for writing Fourier series to describe relations.

    The sine of x may be computed to desired accuracy using sin x = x - x3/3! + x5/5! - x7/7! + ...


    The cosine, cos x, is the ratio of x to r, the ratio of the horizontal component to the radius.

    It is useful in describing natural phenomena particularly in writing Fourier series for relations.

    The cosine of a number, cos(x), may be evaluated to desired accuracy using the expression cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...


    The cosecant, csc x, is the ratio of r to x. It is the reciprocal of the sine.

    Because it is the reciprocal of the sine, when the sine increases the cosecant decreases. When the sine reaches a maximum, the cosecant reaches a minimum.

    When the sine is 0, the cosecant is undefined or infinite in size.


    The secant, sec x, is the reciprocal of the cosine, the ratio of r to x.

    When the cosine is 0, the secant is undefined. When the cosine reaches a relative maximum, the secant is at a relative minimum.


    The tangent is the ratio of y to x, the ratio of the sine to the cosine: tan(x) = sin(x)/cos(x).

    The tangent is very useful in trigonometry.


    The cotangent is the reciprocal of the tangent. It is the ratio of x to y. It is also the ratio of the cosine to the sine: cot(x) = cos(x)/sin(x).


    This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

    You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. YOU MAY NOT MAKE ANY ADDITIONAL COPIES OF THIS PAGE, ITS PICTURE(S), ITS SOUND CLIP(S), OR ITS ANIMATED GIFS WITHOUT PERMISSION.


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