Trigonometry

The fundamental trigonometric functions, Sin and Cos, are defined by ratios of sides of a right-angle triangle, as functions of an angle of the triangle.

In deference to the SI choice of unit of angle, I'll define the more usual functions (whose names I don't capitalise)

for which sin' = cos and cos' = −sin. These dimensionless forms of the trigonometric functions are closely related to the exponential function, defined by exp' = exp, exp(0) = 1; in the complex plane (with i as a square root of −1), this is periodic with period 2.π.i and exp(i.t) = cos(t) +i.sin(t), to which I'll return.

To complete the standard family of trigonometric functions, define the tangent, secant and their complements by:

The prefix co generally means do it to the angle's complement within a right-angle triangle, i.e. coF = F∘(: left−a ←a :), but since Cos is an even function, i.e. Cos = Cos∘(: −a ←a :), one can replace left−a with a−left in some cases (as noted) – though not for Tan and coTan. Some authors abbreviate the co-secant to csc rather than cosec. The usual abbreviation for secant makes possible an obvious if silly joke, potentially suitable for a valentine's day card to a geek.

The studious reader can now use Pythagoras' theorem and the derivatives of Sin and Cos to verify that

and derive suitably negated equivalents for cotan and cosec. It is also possible to express each of the trigonometric function, as computed for an arbitrary angle, in terms of the tangent of half that angle.

The hyperbolic companion functions

As discussed elsewhere, exp(i.t) = cos(t) +i.sin(t), from which we may infer

which will allow us to extend our definitions of cos and sin from the real line to the whole complex plane (by letting t range over the latter, rather than restricting it to the former). Analogy with the usual trigonometric functions encourages introduction of the so-called hyperbolic companion functions (so-called because their role in hyperbolae is analogous to that of the trigonometric functions in circles):

for which cosh' = sinh, sinh' = cosh, with sinh(0) = 0 and cosh(0) = 1, while cosh(t).cosh(t) = 1 +sinh(t).sinh(t) for all t. Presumably cosech can be called csch, too.

Etymological digression

The names of the functions all come from constructions involving a circle and a right-angle triangle. In each case, if the circle's radius is taken as unit of length, the function is the name of a line in the construction: otherwise, it is the ratio of this line's length to the radius.

A line which touches a circle without crossing it is called a tangent (or tangent line) from the Latin for touching. If we draw a radius of the circle through the point at which our tangent touches the circle, the radius and tangent meet in a right angle. If we construct a second radius, making angle a with the one to the touching-point, and extend this new radius beyond the circle until it meets the tangent line, we obtain a right angle triangle. One side of this triangle is an interval of the tangent line; its length is Tan(a) times the circle's radius. The hypotenuse of this triangle is known as a secant (or secant line) and is likewise Sec(a) times the circle's radius. I gather the word secant means cutting or perhaps cutting the circle twice or in two (it's one half of a diameter) but don't know enough Latin.

When a straight line cuts a circle, neither tangent nor through the center, the part of the line inside the circle is known as a chord. If we join each end of the chord to the centre of the circle we get a triangle; two of its sides are radii, so of equal length; the chord is said to subtend the angle in which these two radii meet. If we draw a radius of the circle through the centre of the chord, the two meet in a right angle (the case of a tangent, above, is just the limiting case of a chord of zero length); this radius also bisects the angle subtended by the chord. Hindu mathematicians thus spoke of the half-chord, or ardha-jiva, as a function of the half-angle between the bisecting radius and either end-radius of the chord. (Larry Gonick, in volume III of his Cartoon History of the Universe, translates the Hindu term as bow-string.) This got shortened to jiva, transcribed into Arabic, mis-read by some Europeans (in the 1300s) as an Arabic word with different vowels, translated to Latin as sinus and mangled into the word sine. (The history of words is full of craziness ;^)

The word trigonometry comes from the Greek for measuring triangles.

Everything in terms of Tan of half the angle

The Sin and Cos functions encode Pythagoras' theorem as the assertion that the sum of their squares is one; and their addition formulae naturally enable us to infer their values at multiples of an angle, given their values at that angle. Consequently, given an angle a:

Sec(a).Sec(a)
= 1/Cos(a)/Cos(a)
= (Cos(a).Cos(a) +Sin(a).Sin(a))/Cos(a)/Cos(a)
= 1 +Tan(a).Tan(a), whence
Cos(2.a)
= Cos(a).Cos(a) −Sin(a).Sin(a)
= Cos(a).Cos(a).(1 −Tan(a).tan(a))
= (1 −Tan(a).Tan(a)) / (1 +Tan(a).Tan(a))
Sin(2.a)
= 2.Sin(a).Cos(a)
= 2.Tan(a).Cos(a).Cos(a)
= 2.Tan(a) / (1 +Tan(a).Tan(a))
Tan(2.a)
= Sin(2.a) / Cos(2.a)
= 2.Tan(a) / (1 −Tan(a).Tan(a))

so that we are able to express all trigonometric functions (obviously we get Sec, coSec and coTan by inverting the above) of any given angle as functions of the tangent of half that angle.

We can do similar for the hyperbolic cousins, now with a simply real:

sech(a).sech(a)
= 1/cosh(a)/cosh(a)
= (cosh(a).cosh(a) −sinh(a).sinh(a))/cosh(a)/cosh(a)
= 1 −tanh(a).tanh(a), whence
cosh(2.a)
= cosh(a).cosh(a) +sinh(a).sinh(a)
= (1 +tanh(a).tanh(a))/(1 −tanh(a).tanh(a))
sinh(2.a)
= 2.sinh(a).cosh(a)
= 2.tanh(a)/(1 −tanh(a).tanh(a))
tanh(2.a)
= sinh(2.a) / cosh(2.a)
= 2.tanh(a)/(1 +tanh(a).tanh(a))

which is exactly the same aside from reversing the sing of the tan2 terms while making the functions hyperbolic.


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