Curcular Definition of Trigonometric Functions

Trigonometric functions are first defined for acute (less than the right) angles such that may be encountered in right triangles. If abc and a'b'c' are two similar right triangles, then, say, a/c = a'/c'. So that the ratio a/c of a leg to the hypotenuse is a shape property. The shape of a right triangle is fully defined by one of its acute angles. It follows that the ratio a/c depends only on one of the acute angles of the triangle. It is customary to define

  a/c = sin α = cos β

and similarly,

  b/c = sin β = cos α,

where sin (reads sine) and cos (reads cosine) are the most fundamental trigonometric functions. In addition we have

tangent:tan α= a/b
cotangent:cot α= b/a
secant:sec α= c/b
cosecant:csc α= c/a
versine:versin α= (c - b) / c

and there are more.

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As can be seen, the functions are defined not just for acute angles. The above definitions are extended to make sine and cosine periodic with period 2π:

sin (α + 2π)= sin α
cos (α + 2π)= cos α.

The extensions are made according to the following rules:

sin (α + π/2)= cos α
cos (α + π/2)= - sin α
sin (α + π)= - sin α
cos (α + π)= - cos α.

As a consequence, tangent and cotangent aquire a period of π:

tan (α + π)= tan α
cot (α + π)= cot α.

Sine is set to be odd:

sin (-α)= - sin α

Cosine is even:

cos (-α)= cos α.

This make tangent and cotangent odd.

Versine is defined as

versin (-α)= 1 - cos α.

This function is used only rarely. However we did stumble on an example where it proved useful.

Secant and cosecant are the reciprocals of cosine and sine, respectively:

sec α= 1 / cos α
csc α= 1 / sin α.

The derivatives of sine and cosine are among the first obtained in calculus:

sin' α= cos α
cos' α= - sin α.

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