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# 15. Trigonometric

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## 15.1 Introduction to Trigonometric

Maxima has many trigonometric functions defined. Not all trigonometric identities are programmed, but it is possible for the user to add many of them using the pattern matching capabilities of the system. The trigonometric functions defined in Maxima are: `acos`, `acosh`, `acot`, `acoth`, `acsc`, `acsch`, `asec`, `asech`, `asin`, `asinh`, `atan`, `atanh`, `cos`, `cosh`, `cot`, `coth`, `csc`, `csch`, `sec`, `sech`, `sin`, `sinh`, `tan`, and `tanh`. There are a number of commands especially for handling trigonometric functions, see `trigexpand`, `trigreduce`, and the switch `trigsign`. Two share packages extend the simplification rules built into Maxima, `ntrig` and `atrig1`. Do `describe(command)` for details.

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## 15.2 Definitions for Trigonometric

Function: acos (x)

- Arc Cosine.

Function: acosh (x)

- Hyperbolic Arc Cosine.

Function: acot (x)

- Arc Cotangent.

Function: acoth (x)

- Hyperbolic Arc Cotangent.

Function: acsc (x)

- Arc Cosecant.

Function: acsch (x)

- Hyperbolic Arc Cosecant.

Function: asec (x)

- Arc Secant.

Function: asech (x)

- Hyperbolic Arc Secant.

Function: asin (x)

- Arc Sine.

Function: asinh (x)

- Hyperbolic Arc Sine.

Function: atan (x)

- Arc Tangent.

Function: atan2 (y, x)

- yields the value of `atan(y/x)` in the interval `-%pi` to `%pi`.

Function: atanh (x)

- Hyperbolic Arc Tangent.

Package: atrig1

The `atrig1` package contains several additional simplification rules for inverse trigonometric functions. Together with rules already known to Maxima, the following angles are fully implemented: `0`, `%pi/6`, `%pi/4`, `%pi/3`, and `%pi/2`. Corresponding angles in the other three quadrants are also available. Do `load(atrig1);` to use them.

Function: cos (x)

- Cosine.

Function: cosh (x)

- Hyperbolic Cosine.

Function: cot (x)

- Cotangent.

Function: coth (x)

- Hyperbolic Cotangent.

Function: csc (x)

- Cosecant.

Function: csch (x)

- Hyperbolic Cosecant.

Option variable: halfangles

Default value: `false`

When `halfangles` is `true`, half-angles are simplified away.

Package: ntrig

The `ntrig` package contains a set of simplification rules that are used to simplify trigonometric function whose arguments are of the form `f(n %pi/10)` where f is any of the functions `sin`, `cos`, `tan`, `csc`, `sec` and `cot`.

Function: sec (x)

- Secant.

Function: sech (x)

- Hyperbolic Secant.

Function: sin (x)

- Sine.

Function: sinh (x)

- Hyperbolic Sine.

Function: tan (x)

- Tangent.

Function: tanh (x)

- Hyperbolic Tangent.

Function: trigexpand (expr)

Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. For best results, expr should be expanded. To enhance user control of simplification, this function expands only one level at a time, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the switch `trigexpand: true`.

`trigexpand` is governed by the following global flags:

`trigexpand`

If `true` causes expansion of all expressions containing sin's and cos's occurring subsequently.

`halfangles`

If `true` causes half-angles to be simplified away.

`trigexpandplus`

Controls the "sum" rule for `trigexpand`, expansion of sums (e.g. `sin(x + y)`) will take place only if `trigexpandplus` is `true`.

`trigexpandtimes`

Controls the "product" rule for `trigexpand`, expansion of products (e.g. `sin(2 x)`) will take place only if `trigexpandtimes` is `true`.

Examples:

 ```(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand; 2 2 (%o1) - sin (x) + 3 cos (x) + x (%i2) trigexpand(sin(10*x+y)); (%o2) cos(10 x) sin(y) + sin(10 x) cos(y) ```
Option variable: trigexpandplus

Default value: `true`

`trigexpandplus` controls the "sum" rule for `trigexpand`. Thus, when the `trigexpand` command is used or the `trigexpand` switch set to `true`, expansion of sums (e.g. `sin(x+y))` will take place only if `trigexpandplus` is `true`.

Option variable: trigexpandtimes

Default value: `true`

`trigexpandtimes` controls the "product" rule for `trigexpand`. Thus, when the `trigexpand` command is used or the `trigexpand` switch set to `true`, expansion of products (e.g. `sin(2*x)`) will take place only if `trigexpandtimes` is `true`.

Option variable: triginverses

Default value: `all`

`triginverses` controls the simplification of the composition of trigonometric and hyperbolic functions with their inverse functions.

If `all`, both e.g. `atan(tan(x))` and `tan(atan(x))` simplify to x.

If `true`, the `arcfun(fun(x))` simplification is turned off.

If `false`, both the `arcfun(fun(x))` and `fun(arcfun(x))` simplifications are turned off.

Function: trigreduce (expr, x)
Function: trigreduce (expr)

Combines products and powers of trigonometric and hyperbolic sin's and cos's of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.

See also `poissimp`.

 ```(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x); cos(2 x) cos(2 x) 1 1 (%o1) -------- + 3 (-------- + -) + x - - 2 2 2 2 ```

The trigonometric simplification routines will use declared information in some simple cases. Declarations about variables are used as follows, e.g.

 ```(%i1) declare(j, integer, e, even, o, odd)\$ (%i2) sin(x + (e + 1/2)*%pi); (%o2) cos(x) (%i3) sin(x + (o + 1/2)*%pi); (%o3) - cos(x) ```
Option variable: trigsign

Default value: `true`

When `trigsign` is `true`, it permits simplification of negative arguments to trigonometric functions. E.g., `sin(-x)` will become `-sin(x)` only if `trigsign` is `true`.

Function: trigsimp (expr)

Employs the identities sin(x)^2 + cos(x)^2 = 1 and cosh(x)^2 - sinh(x)^2 = 1 to simplify expressions containing `tan`, `sec`, etc., to `sin`, `cos`, `sinh`, `cosh`.

`trigreduce`, `ratsimp`, and `radcan` may be able to further simplify the result.

`demo ("trgsmp.dem")` displays some examples of `trigsimp`.

Function: trigrat (expr)

Gives a canonical simplifyed quasilinear form of a trigonometrical expression; expr is a rational fraction of several `sin`, `cos` or `tan`, the arguments of them are linear forms in some variables (or kernels) and `%pi/n` (n integer) with integer coefficients. The result is a simplified fraction with numerator and denominator linear in `sin` and `cos`. Thus `trigrat` linearize always when it is possible.

 ```(%i1) trigrat(sin(3*a)/sin(a+%pi/3)); (%o1) sqrt(3) sin(2 a) + cos(2 a) - 1 ```

The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.

 ```(%i1) c: %pi/3 - a - b; %pi (%o1) - b - a + --- 3 (%i2) bc: sin(a)*sin(3*c)/sin(a+b); sin(a) sin(3 b + 3 a) (%o2) --------------------- sin(b + a) (%i3) ba: bc, c=a, a=c\$ (%i4) ac2: ba^2 + bc^2 - 2*bc*ba*cos(b); 2 2 sin (a) sin (3 b + 3 a) (%o4) ----------------------- 2 sin (b + a) %pi 2 sin(a) sin(3 a) cos(b) sin(b + a - ---) sin(3 b + 3 a) 3 - -------------------------------------------------------- %pi sin(a - ---) sin(b + a) 3 2 2 %pi sin (3 a) sin (b + a - ---) 3 + --------------------------- 2 %pi sin (a - ---) 3 (%i5) trigrat (ac2); (%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a) - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a) - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a) + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a) + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b) + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a) - 9)/4 ```

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