Circular Trigonometric Functions

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 the Complex Plane, the nine Circular Trigonometric functions are defined as follows:

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

That the two definitions of the haversed sine are equivalent
follows from a sneak preview of the double-angle formula for the
cosine. The first definition is the one that motivates the name;
but it entails excessive round-off error.
The sine, tangent, cotangent, cosecent, and coversed sine --
by inheritance -- are odd functions, while the cosine, secant,
versed sine, and haversed sine are even.
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 listing. 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.
sin(x + y + z) = sin(x) cos(y) cos(z) + cos(x) sin(y) cos(z) + cos(x) cos(y) sin(z) - sin(x) sin(y) sin(z)
cos(x + y + z) = cos(x) cos(y) cos(z) - cos(x) sin(y) sin(z) - sin(x) cos(y) sin(z) - sin(x) sin(y)
cos(z)
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 Differences
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 complex addition 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) = x + i y; x = Arccos(sqrt(((1 - (u^2 +
v^)) - w) / 2)) = Arcsin(sqrt(((1 + (u^2 + v^2)) + w) /
2)), y = Arccosh(sqrt(((1 + (u^2 + v^2)) + w) / 2)) =
Arcsinh(sqrt((- (1 - (u^2 + v^2)) + w) / 2)); w =
sqrt((u2 + v^2)^2 - 2 (u^2 - v^2) + 1)
Arccos(u + i v) = x + i y; x = Arccos(sqrt(((1 + (u^2 -
v^2)) + w) / 2)) = Arcsin(sqrt(((1 - (u^2 - v^2)) - w) /
2)), y = Arccosh(sqrt(((1 + (u^2 - v^2)) - w) / 2)) =
Arcsinh(sqrt((- (1 - (u^2 - v^2)) - w) / 2)); w =
sqrt(u^2 - v^2)^2 - s (u^2 + v^2) + 1) 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((1 - cos(x)) / 2)
cos(x / 2) = +- sqrt((1 + cos(x)) / 2)
tan(x / 2) =
= +- sqrt((1 - cos(x)) / (1 + cos(x)))
= sin(x) / (1 + cos(x))
= (1 - cos(x)) / sin(x)
cot(x / 2) = 1 / tan(x / 2) =

= +- sqrt((1 + cos(x)) / (1 - cos(x)))
= (1 + cos(x)) / sin(x)
= sin(x) / (1 - cos(x))
The half-angle formulae for the circular secant, cosecant, versed sine, coversed sine, and
haversed sine are not interesting. Multiple-Angle Formulae
From
exp(i x) = cos(x) + i sin(x)
it follows that, for any natural number n,
exp(i n x) = cos(n x) + i sin(n x) = (cos(x) + i sin(x))^n
By the binomial formula, expansion of the right-most expression yields the
multiple-angle formulae for the circular-trigonometric functions
cos(n x) = (cos(x))^n - n! / ((n - 2)! 2!) (cos(x))^(n - 2) (sin(x))^2 +
n! / ((n - 4)! 4!) (cos(x))^(n - 4) (sin(x))^4 -+....
sin(n x) = (sin(x))^n - n! / ((n - 2)! 2!) (sin(x))^(n - 2) (cos(x))^2 +
n! / ((n - 4)! 4!) (sin(x))^(n - 4) (cos(x))^4 -+.... 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 parameterization of an ellipse -- a circle would
be obtained by making b equal to a.
For conic sections, please see conic.
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
By l'Hospital's rule, it follows that the limit, as x approaches zero, of
sin(x) / x is one.
Inverse
Let x = sin(y) and differentiate it to obtain dx / dy =
cos(y). Employ the appropriate identity 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 = - ln(sec(x) - tan(x)) + C
int(csc(x)) = ln(scs(x) - cot(x)) + C = - 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:
Take the infinite Geometric series
1 / (1 + x) = 1 - x + x^2 - x^3 + ... + (- 1)^n x^n +
....
for any x, in Complex, whose the absolute value is less than
one. Replace x by x^2 and integrate to obtain the McLaurin's
series for the circular arctangent.
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, secant, cosecant, versed sine, and coversed
sine are not interesting.
haversine hav(x) = (1 / 2) (x^2 / 2 - x^4 / 24 + ... + (-
1)^n x^(2 n + 2) / ((2 n + 2)!) + ....
arctangent Arctan(x) = x - x^3 / 3 + x^5 / 5 + ... + (-
1)^n x^(2 n + 1) / (2 n + 1) + .... provided that abs(x) <
1.
It is obtained by integration of the geometric series 1 / (1 + x^2) = 1
- x^2 + x^4 - x^6 + ... + (- 1)^n x^(2 n) + .... provided that abs(x) <
1.
The McLaurin's series formulae for the circular arc sine,
arc cosine, arc cotangent, arc secant, arc cosecant, arc
versed sine, arc coversed sine, and arc haversed sine are
not interesting. The values of the inverse circular trigonometric functions have to be obtained from that of
the foregoing arctangent, by solving the quadratic
equations of the identities and definitions.
Arcsin(x) = Arctan(x / sqrt((1 - x)(1 + x)))
Arccos(x) = Arctan(sqrt((1 - x)(1 + x)) / x)
Arccot(x) = Arctan(1 / x)
Arcsec(x) = Arctan(sqrt((x - 1)(x + 1)))
Arccsc(x) = Arctan(1 / sqrt((x - 1)(x + 1))) Infinite Products
From the theorem which states that any function without zeros
or poles is a constant, we may obtain the infinite products of a
function. The infinite product expansion of the sine or cosine
functions converges too slowly to be practical for numerical
calculation.
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, secant, cosecant, versed sine, coversed sine,
and haversed sine are not interesting.
A table of some frequently used values of the circular
trigonometric functions is provided.
Hyperbolic Trigonometric Functions
The purely imaginary counterpart of the Circular Trigonometric
functions is called the Hyperbolic
Trigonometric functions.
Copywrite © 1997,8,9, 2000 R. I. 'Scibor-Marchocki last modified on Sunday
24-th September 2000.