TRIGONOMETRIC FUNCTIONS AND EQUIVALENTS
(TRIGFUNC.TXT)
This is a summary of numerous functions and their equivalents
necessary when working with the language limitations of BASIC
and others which do not include pre-programmed functions beyond
sine, cosine, tangent, arctangent and others.
Many are uncommon and seldom encountered but nevertheless
valuable under specialized conditions. Indeed, some solutions
are rarely mentioned in most references.
Implementations here are presented in typical BASIC format but
are readily translated to any other keeping in mind the
nuances of your working language. A final section sets forth
a few programming hints, and tips which help avoid run-time
errors.
I - Common Functions
--------------------
Secant SEC(X) = 1 / COS(X)
Cosecant CSC(X) = 1 / SIN(X)
Cotangent COT(X) = 1 / TAN(X)
Inverse Sine ARCSIN(X) = ATN(X / SQR(1-X*X))
Inverse Cosine ARCCOS(X) = - ATN(X / SQR(1-X*X)) + PI/2
Inverse Secant ARCSEC(X) = ATN(SQR(X*X-1)) + (SGN(X) -1) * PI/2
Inverse Cosecant ARCCSC(X) = ATN(1 / SQR(X*X-1)) + (SGN(X) -1) * PI/2
Inverse Cotangent ARCCOT(X) = PI/2 - ATN(X) | or | PI/2 + ATN(-X)
II - Hyperbolic Functions
-------------------------
Sine SINH(X) = (EXP(X) - EXP(-X)) / 2
Cosine COSH(X) = (EXP(X) + EXP(-X)) / 2
Tangent TANH(X) = -2 * EXP(-X) / (EXP(X) + EXP(-X)) + 1
Secant SECH(X) = 2 / (EXP(X) + EXP(-X))
Cosecant CSCH(X) = 2 / (EXP(X) - EXP(-X))
Cotangent COTH(X) = 2 * EXP(-X) / (EXP(X) - EXP(-X)) + 1
Inverse Sine ARCSINH(X) = LOG(X + SQR(X*X+1))
Inverse Cosine ARCCOSH(X) = LOG(X + SQR(X*X-1))
Inverse Tangent ARCTANH(X) = LOG((1+X) / (1-X)) / 2
Inverse Secant ARCSECH(X) = LOG((1+SQR(1-X*X)) / X)
Inverse Cosecant ARCCSCH(X) = LOG((SGN(X) * SQR(X*X+1) + 1) / X)
Inverse Cotangent ARCCOTH(X) = LOG((X+1) / (X-1)) / 2
III - Hints and Tips
--------------------
The comments related here apply specifically to QuickBasic v4.0 and
lower and MS GWBasic, unless otherwise mentioned. Other Basic
implementations may be better or worse in certain idiosyncrasies
which should be determined by users before placing total faith in any
suggested anomoly trapping.
These fundamental needs should be acceptable in all cases:
PI = 4 * ATN(1)
J = PI / 180
J is a facilitiation constant for Degrees - Radians - Degrees
conversion, thusly:
Variable (Xd) * J = Variable (Xr) ..... degrees to rads
Variable (Xr) / J = Variable (Xd) ..... rads to degrees
The question of single or double-precision use may be one of importance
to your application. With MS QB, final precision deteriorates significantly
when the above trig transformations are employed. In other words, don't expect
single precision (7-8 digits) when using s-p or 15-16 digits in d-p.
Depending on vector position, the end result can be as poor as 1/2 the
expected accuracy. Also, if one wants highest accuracy, it is
essential that low digit numerics (eg: 0.815) be declared as d-p
(.815#) otherwise they will be treated as single precision even though
attached in a series to a variable which has been declared as d-p.
CAUTIONS
--------
1. Entering arguments at or very near ñ1 can produce attempts to
divide by zero. Some form of trapping is essential to avoid program
stoppage. Appropriate filtering with bypasses, alternate paths
are suggested.
2. In complex trigometric manipulations it is possible for the end
result to exceed ñ1. This may be due to binary quirks or
simply erroneous procedures. Some form of trapping is also
needed to avoid crashes when such are used as entering arguments.
3. Similarly, trapping is required for 0 and -X values when entering
some functions using LOG. Also note that a few functions with SQR
have other invalid ranges for entering arguments.
IV - Acknowledgement
Source material for Sections I and II from Texas Instruments,
TI Extended BASIC handbook for the TI-99/4, 1981.
--------------------
Prepared by Anthony W. Severdia : San Francisco, CA : FEB 1989
This file is in Public Domain
--------------------
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