TRIGONOMETRIC FUNCTIONS AND EQUIVALENTS

 

                    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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