Circular Trigonometric Functions

Each of the circular trigonometric functions inherits from the exponential function 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 secant sec(x) = 1 / cos(x)
The cosecant 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 = (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 - 2 (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 -+.... <<<< Wrong!

Shame on all of you. The foregoing formula for the sine is wrong and nobody noticed. I just wanted to use it for sin(3 x), on the 2-nd of March 2002, and discovered the error. :-)
Let us write these formulae more concisely by employing the binomial coefficients

C(n, m) = n! / ((n - m)! m!).

Then, the two formulae become

cos(n n) = (cos(x))^n - C(n, 2) (cos(x))^(n - 2) (sin(x))^2 + ... + C(n, 2 m) (cos(x))^(n - 2 m) (- (sin(x))^2)^m + ....
sin(n x) = sin(x) (C(n, 1) (cos(x))^(n - 1) - C(n, 3) (sin(x))^2 (cos(x))^(n - 3) + ... + C(n, 2 m + 1) (- (sin(x))^2)^m (cos(x))^(n - (2 m + 1)) + ...).

Of course, these series terminate before the exponents become negative. I hope that I have these formulae correct this second try. :-)

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

MacLaurin's Series

For any x in Complex, we have the following MacLaurin'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 MacLaurin'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 MacLaurin'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 MacLaurin'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)))

Principal value of the inverse functions

The choice of the principal value of each of the foregoing inverse functions is a mater of convention. For a non-negative x, everybody agrees that the principal value of each of these functions is in the closed interval [0, pi / 2]. :For negative x, mostly is is agreed that the inverse sine is in [- pi / 2, 0) and the inverse cosine is in (pi / 2, pi}. However, textbooks differ for the remaining functions. Calculus textbooks tend to favor whatever it takes to preserve the foregoing formulae for the derivatives of the inverse functions. For a different view on the principal value of the inverse tangent, see its discussion under the logarithm of a complex number.

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.