Circular Trigonometric Functions

Each of the circular trigonometric functions has a modulus of periodicity of 2 pi, except that the tangent and cotanget have a modulus of periodicity of pi.

Definitions

We desire to have an explicit and easy set of functions for the purely imaginary counterpart of the Hyperbolic Trigonometric Functions. Thus, we define the circular sine and cosine as the purely imaginary counterparts of the corresponding hyperbolic functions. The remaining seven functions, we define in a manner analogous to that of the corresponding hyperbolic functions.

For an x in Complex, we define the nine Circular Trigonometric functions as follows:

The sine sin(x) = sinh(i x) / i
The cosine cos(x) = cosh(i x)
The tangent tan(x) = sin(x) / cos(x)
The cotangent cot(x) = 1 / tan(x)
The secent sec(x) = 1 / cos(x)
The cosecent csc(x) = 1 / sin(x)
The versed sine versin(x) = 1 - cos(x)
The coversed sine coversin(x) = 1 - sin(x)
The haversed sine haversin(x) = (1 - cos(x)) / 2

It follows that, in terms of the corresponding hyperbolic function, the last seven are:

sin(x) = sinh(i x) / i
cos(x) = cosh(i x)
tan(x) = tanh(i x) / i
cot(x) = i coth(i x)
sec(x) = sech(i x)
csc(x) = i csch(i x)
versin(x) = 1 - cosh(i x)
coversin(x) = 1 - sinh(i x) / i
haversin(x) = (1 - cosh(i x)) / 2

We have repeated the first two for completeness of this listsing. The remainder of equations in this summary of the Circular Trigonometric Functions may be derived by the substitution from the foregoing list into the corresponding equations of the Hyperbolic Trigonometric Functions.

Identities

For any x in the Cartesian product of Complex by Complex, we have the following identities:

(sin(x))^2 + (cos(x))^2 = 1
(tan(x))^2 + 1 = (sec(x))^2
(cot(x))^2 + 1 = (csc(x))^2

Addition Theorems

Real

For any (x, y) in the Cartesian product of Complex by Complex, we have the following real addition theorems:

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) tan(y))
cot(x + y) = (1 - cot(x) cot(y)) / ( cot(x) + cot(y))
The addition theorems for the circular secent, cosecent, versed sine, coversed sine, and haversed sine are not interesting.

Products

For any (x, y) in the Cartesian product of Complex by Complex, we have the following real product theorems

sin(x) cos(y) = (sin(x + y) + sin(x - y)) / 2
cos(x) cos(y) = (cox(x + y) + cos(x - y)) / 2
sin(x) sin(y) = (cos(x - y) - cos(x + y)) / 2

Sums or Differences

Let u = x + y and v = x - y. Substitution in the foregoing three equations yields the Sums or Differencess

sin(u) + sin(v) = 2 sin((u + v) / 2) cos((u - v) / 2)
cos(u) + cos(v) = 2 cos(u + v) / 2) cos(u - v) / 2)
cos(v) - cos(u) = 2 sin((u + v) / 2) sin((u - v) / 2)

Complex

For any (x, y) in the Cartesian product of Complex by Complex, we have the following comples addtition theorems:

exp(x + i y) = exp(x) (cos(y) + i sin(y)).
sin(x + i y) = sin(x) cosh(y) + i cos(x) sinh(y).
cos(x + i y) = cos(x) cosh(y) - i sin(x) sinh(y).
tan(x + i y) = (tan(x) + i tanh(y)) / (1 - i tan(x) tanh(y)).
cot(x + i y) = (cot(x) coth(y) - i) / (i cot(x) + coth(y)).
The complex addition theorems for the circular secent, cosecent, versed sine, coversed sine, and haversed sine are not interesting.

The special cases where x is zero are as follows:

exp(i y) = cos(y) + i sin(y).
sin(i y) = i sinh(y).
cos(i y) = cosh(y).
tan(i y) = i tanh(y).
cot(i y) = - i coth(y).
sec(i y) = sech(y).
csc(i y) = - i csch(y).
versin(i y) = versinh(y).
haversin(i y) = haversinh(y).

On the other hand, we may invert the first three of the foregoing complex addition theorems. Set the right-hand side equal to u + i v. Then collect the real and imaginary parts on one side of the equation; each part has to be zero. Then by employing the identities, we obtain

Arcexp(u + i v) = ln(u + i v) = (1 / 2) ln(u^2 + v^2) + i Arctan(u / v)
Arcsin(u + i v) = ???
Arccos(u + i v) = ???

Double-Angle Formulae

For any x in Complex, we have the following double-angle formulae:

sin(2 x) =2 sin(x) cos(x).
cos(2 x) = (cos(x))^2 - (sin(x))^2 = 1 - 2 (sin(x))^2) = 2 (cos(x))^2 - 1.
tan(2 x) = 2 tan(x) / (1 - (tan(x))^2).
cot(2 x) = (1 - (cot(x))^2) / (2 cot(x)).
sec(2 x) = (sec(x) csc(x))^2 / ((sec(x))^2 + (csc(x))^2).
csc(2 x) = (sec(x) csc(x))^2 / (2 sec(x) csc(x)).
versin(2 x) = 2 (sin(x))^2).
coversin(2 x) = 1 - 2 sin(x) cos(x).
haversin(2 x) = (sin(x))^2).

Half-Angle Formulae

For any x in Complex by Complex, we have the following half-angle formulae:

sin(x / 2) = +- sqrt(cos(x) - 1) / 2
cos(x / 2) = +- sqrt(cos(x) + 1) / 2
tan(x / 2) =
- = +- sqrt((cos(x) - 1) / (cos(x) + 1)).
- = sin(x) / (cos(x) + 1)..
- = (cos(x) - 1) / sin(x).
The half-angle formulae for the circular cotangent, secent, cosecent, versed sine, coverssed sine, and haversed sine are not interesting.

Ellipse

A parametric equation of an ellipse in the Cartesian product of Complex by Complex, is given by

(x, y) = (a cos(t), b sin(t)) for any t in Complex and any constant (a, b), called the semi-axes, in the Cartesian product of Complex by Complex.

Active (that is with time being known)navigation employs ellipses and ellipsoids.

Historically, these functions have been called circular because of this perametrization of an ellipse -- a circle would be obtained by making b equal to a.

Calculus

Derivatives

For any x in Complex by Complex, we have the following derivative formulae:

Direct

d sin(x) / dx = cos(x).
d cos(x) / dx = - sin(x).
d tan(x) / dx = (sec(x))^2.
d cot(x) / dx = - (csc(x))^2.
d sec(x) / dx = tan(x) sec(x).
d csc(x) / dx = - cot(x) csc(x).
d versin(x) / dx = sin(x).
d coversin(x) / dx = - cos(x).
d haversin(x) / dx = sin(x) / 2.

Inverse

Let x = sin(y) and differentiate it to obtain dx / dy = cos(y). Employ the appropriate identities to obtain dx / dy = sqrt(1 - (sin(y))^2). Then dy / dx = 1 / sqrt(1 - (sin(y))^2). Thus, we have obtained the first of the derivative formulae of the inverse circular trigonometric functions

d Arcsin(x) / dx = 1 / sqrt(1 - x^2)
d Arccos(x) / dx = - 1 / sqrt(1 - x^2)
d Arctan(x) / dx = 1 / (1 + x^2)
d Arccot(x) / dx = - 1 / (1 + x^2)
d Arcsec(x) / dx = x sqrt(x^2 - 1)
d Arccsc(x) / dx = - x sqrt(x^2 - 1)

Their primary utility is as antiderivatives.

Integrals

Lacking the glyph for the integral sign, we are going to indicate the definite integral of a function f(x) with respect to x on the interval from a to b as int(f(x), x, a, b); the indefinite integral as int(f(x), x). When the dummy variable of integration is obvious, we will omit it, as being implied. C is the constant of integration. For any x in Complex, we have the following integral formulae:

int(sin(x)) = - cos(x) + C.
int(cos(x)) = sin(x) + C.
int(tan(x)) = - ln(cos(x)) + C = ln(sec(x)) + C.
int(cot(x)) = ln(sin(x)) + C.
int(sec(x)) = ln(sec(x) + tan(x)) + C.
int(csc(x)) = - ln(csc(x) + cot(x)) + C.
int(versin(x)) = x + cos(x) + C.
int(coversin(x)) = x - sin(x) + C.
int(haversin(x)) = (x + cos(x)) / 2 + C.

Infinite Expansions

McLaurin's Series

For any x in Complex, we have the following McLaurin's Series:

sine sin(x) = x - x^3 / 6+ x^5 / 120 + ... + (- 1)^n x^(2 n + 1) / (2 n + 1)! + ....
cosine cos(x) = 1 - x^2 / 2 +x^4 / 24 + ... + (- 1)^n x^(2 n) / (2 n)! + ....
The McLaurin's series formulae for the circular tangent, cotangent, secent, cosecent, versed sine, coverssed sine, and haversed sine are not interesting.
arctangent arctan(x) = x - x^2 / 2 + x^3 / 3 + ... + (- 1)^n x^(n + 1) / (n + 1) + .... provided that abs(x) < 1.
The McLaurin's series formulae for the circular arc sine, arc cosine, arc cotangent, arc secent, arc cosecent, arc versed sine, arc coverssed sine, and arc haversed sine are not interesting. The values of the inverse circular trigonometic functions have to be obtained from that of the foregoing arctangent, by solving the quadratic equations of the identities and definitions.

Infinite Products

From the theorem which states that any function without zeros or poles is a constant, we may obtain the infiite products of a function.

For brevity, let y = (2 x / pi)^2. For any x in Complex, we have the following infinite products

sine sin(x) = (2 x / pi) ((4 - y) / 3)(( 16 - y) / 5) ((36 - y) / 35) ...(((2 n)^2 - y) / ((2 n)^2 - 1)) ....
cosine cos(x) = (1 - y) (9 - y) (25 - y) ... ((2 n + 1)^2 - y) ....
The infinite product formulae for the circular tangent, cotangent, secent, cosecent, versed sine, coverssed sine, and haversed sine are not interesting.

Hyperbolic Trigonometric Functions

The purely imaginary counterpart of the Hyperbolic Trigonometric functions is called the Hyperbolic Trigonometric functions.