20.64 TRIGD: Trigonometrical Functions with Degree Arguments

This module provides facilities for the numerical evaluation and algebraic simplification of expressions involving trigonometrical functions with arguments given in degrees rather than in radians. The degree-valued inverse functions are also provided.

Author: Alan Barnes

20.64.1 Introduction

This module provides facilities for the numerical evaluation and algebraic simplification of expressions involving trigonometrical functions with arguments given in degrees rather than in radians. The degree-valued inverse functions are also provided.

Any user at all familiar with the normal trig functions in REDUCE should have no trouble in using the facilities of this module. The names of the degree-based functions are those of the normal trig functions with the letter d appended, for example sind, cosd and tand denote the sine, cosine and tangent repectively and their corresponding inverse functions are asind, acosd and atand. The secant, cosecant and cotangent functions and their inverses are also supported and, indeed, are treated more as first class objects than their corresponding radian-based functions which are often converted to expressions involving sine and cosine by some of the standard REDUCE simplifications rules.

Below I give a brief description of the facilities available together with a few examples of their use. More examples and the output that they should produce may be found in the test files trigd-num.tst and trigd-simp.tst and their corresponding log files with extension .rlg which may be found in the directory packages/misc of the REDUCE distribution along with the source code of the module.

These degree-based functions are probably best regarded as functions defined for real values only, but complex arguments are supported for completeness. The numerical evaluation routines are fairly comprehensive for both real and complex arguments. However, few simplifications occur for trigd functions with complex arguments.

The range of the principal values returned by the inverse functions is consistent with those of the corresponding radian-valued functions. More precisely, for asind, atand and acscd the (closure of the) range is \([-90, 90]\) whilst for acosd, acotd and asecd the (closure of the) range is \([0, 180]\). In addition the operator atan2d is the degree valued version of the two argument inverse tangent function which returns an angle in the half-open interval \((-180, 180]\) in the correct quadrant depending on the signs of its two arguments. For \(x>0\), atan2d(y, x) returns the same numerical value as atand(y/x). If \(x=0\) then \(\pm 90\) is returned depending on the sign of \(y\).

It might be thought that the facilities provided in this module could be easily provided by defining suitable rule lists to convert between the radian and degree-based versions of the trig functions. For example:

1: operator sind, asind$
2:  d2r_list := {sind(~x) =>
        sin(x*pi/180), asind(~x) => 180*asin(x)/pi}$
3:  r2d_list := {sin(~x) =>
       sind(180x/pi), asin(~x) => pi*sind(x)/180}$
4:  sind(x+360) where d2r_list$
5:  ws where r2d_list;
         sind(x)
6:  sind(360) where d2r_list;
       0

However, this approach seldom works — try it! The result produced by step 4 defeats the current rule⁵¹ used to simplify expressions of the form \(\sin (x+2\pi )\) although it does manage step 6. The rule list approach is more reliable if differentiation, integration or numerical evaluation of expressions involving sind etc. is required. However it is not particularly convenient even if the rules and operator declarations are stored in a file so that they may be loaded at will.

This module aims to overcome these deficiences by providing the degree-based trig functions as first class objects of the system just like their radian-based cousins. The aim is to provide facilities for numerical evaluation, symbolic simplification and differentiation totally analgous to those for the the basic trig functions and their inverses. It is hoped that the module will be of value to students and teachers at secondary school level as well as being sufficiently powerful and flexible to be of genuine utility in fields where angles measured in degrees (and arc minutes and seconds) are in common usage. For more advanced situations (involving integration, complex arguments and values etc.), users are urged to use the standard trig functions already provided by the system.

20.64.2 Simplification

As in other parts of REDUCE, basic simplification of expressions involving the trigd functions takes place automatically (bracketted terms are multiplied out, like terms are gathered together, zero terms removed from sums and so on). The system knows and automatically applies the basic properties of the functions to simplify the input. For example sind(0) is replaced by 0 and sind(-X) by sind(X). If the switch rounded is off all arithmetic is exact and transcendental functions such as sind are not evaluated numerically even if their arguments are purely numerical.

The built-in simplification rules are totally analogous to those of the standard trig functions namely:

• Replacement of a function application by its value if a simple analytical value is known. For example cosd(60) => 1/2 and acscd(1) => 90. Currently the only argument values where simplification takes place correspond to angles that are integral multiples of \(15^\circ \).

• Use of the odd and even properties of the trig. functions so that for example sind(-x) => -sind(x), cosd(-x) => cosd(x) and
  acosd(-x) => 180 - acosd(x).

• Argument shifts by integral multiples of \(180^\circ \) so that any residual numerical argument lies in the range \(-90^\circ \ldots 90^\circ \). Thus

           sind(x+540) => -sind(x),
           cosd(x+350) => cosd(x-10).
  

• Removal of argument shifts of \(\pm 90^\circ \) so that for example
  sind(x-90) => -cosd(x) and cotd(x+90) => -tand(x).

• Replacement of tand(x) by sind(x)/cosd(x) and e.g., secd(x) by 1/cosd(x) and the like, but only when the final result is simpler than the original.

• Basic properties relating a function and it inverse so that for example
  sind(asind(x)) => x.

• A few basic rules for atan2d when the signs of its arguments can be determined. For example atan2d(y, 0) is replaced by \(\pm 90\) depending on the sign of \(y\).

Extra rules can be added by the user for example addition formulae, double angle rules and tangent half-angle formulae as and when required as described in chapter 11.

Rules are provided for the symbolic differentiation of all the trig functions and their inverses. These rules are sufficient fot the power series of the trig functions and their inverses to be found using either the TPS or TAYLOR packages in the standard way.

20.64.3 Numerical Evaluation

When the switch rounded is on and the arguments of the operators evaluate to numbers, then the floating point value of the expression is calculated to the currently specified precision in the normal way. The bigfloat capabilities are the same as for the standard trig functions.

If these functions are supplied with complex numerical arguments, numerical evaluation will NOT be performed when the switch rounded is on, but the switch complex is off — the input expression will be returned basically unaltered. Similarly inputs such as asind(2) or asecd(0.5) are not evaluated numerically. The values of these expressions are, of course, complex.

If the switch complex is also on , numerical evaluation is performed. For example:

1: load_package trigd$

2:  on rounded;

3:  asecd(2);

         60.0

4:  asecd(0.5);

      asecd(0.5)

5: on complex;

    *** Domain mode rounded changed to complex-rounded

6:  asecd(0.5);

     75.4561292902*i

The function atan2d (like atan2) is only defined if BOTH its arguments are real. If they are also numerical, it will be evaluated whenever rounded is on. Attempting to evaluate it with complex numerical arguments will cause either the unaltered expression to be returned or an error to be raised when the switch complex is off or on respectively.

Note the sine of an angle specified in degrees, minutes and seconds cannot be calculated by calling sind directly with a dms list (i.e. as a list of length 3). Instead one must first convert the dms values to degrees using a call to dms2deg and then call sind on the result. Applied directly to a list (of any length) any trigd function wil be applied to each member of the list separately just like most other REDUCE operators. Here is an example illustrating tese points:

1: load_package trigd$

2: on rounded;

3: sind dms2deg {60, 45, 30};

0.872567064923

4: sind {60,45, 30};

{0.866025403784,0.707106781187,0.5}

5: off rounded;

6: sind{60, 45, 30};

  sqrt(3)   sqrt(2)   1
{---------,---------,---}
     2         2      2

Of course the results will be formatted much more attractively on a terminal supporting nice graphics.

20.64.4 Bugs, Restrictions and Planned Extensions

The behaviour of the numerical evaluation routines for inverse trig functions with complex arguments at branch points could be improved; these values are undefined and attempting to evaluate such a function at one of its branch points ought to raise an error, however sometimes the input expression will be returned unaltered. It is hoped to improve this behaviour in due course.

Currently there are no facilities analogous to those provided in the module TRIGSIMP for the standard trig. functions. There users have a wide range of standard simplification formulae available for use and can control which are to be used depending on the requirements of their particular application: whether to eliminate sin in favour of cos or vice-versa or to get rid of both in favour of tan of half-angles; or whether to use the trigonometrical addition formulae in order to transform trig functions whose arguments are sums into a form where the arguments are single terms or whether to perform the inverse transformations. It is hoped to make the TRIGSIMP faciliites available for use with the TRIGD functions in the near future.

Integration is not directly supported although the approach using rule-lists to convert the TRIGD functions to standard trig ones should work well. Introducing direct supp<ort for integration will not therefore be a priority.

For the standard sine function there is a rule for imaginary arguments namely: sin(i*x) => i*sinh(x). The corresponding rule for the degree version is sind(i*x) => i*sinh(x*pi/180). However, currently such rules are NOT implemented by the system. They may be implemented in future, but it is not a high priority as it is felt that the radian-based trig functions are best suited for such symbolic calculations.

There are NO D versions of the hyperbolic functions — that would be a step too far! And should the new functions be called sinhd and so on? Or perhaps sindh⁵² etc?

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