10.5.2.3 Explicit Simplifications Using Identities

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)

Categories: Trigonometric functions · Simplification functions ·

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.

Categories: Trigonometric functions · Simplification flags and variables ·

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.

Categories: Trigonometric functions · Simplification flags and variables ·

Option variable: triginverses ¶

Default value: true

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.

Categories: Trigonometric functions · Simplification flags and variables ·

Function: trigreduce
    trigreduce (expr, x)
    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

Categories: Trigonometric functions · Simplification functions ·

Function: trigsimp (expr) ¶

Employs the identities \(\sin\left(x\right)^2 + \cos\left(x\right)^2 = 1\) and \(\cosh\left(x\right)^2 - \sinh\left(x\right)^2 = 1\) 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.

Categories: Trigonometric functions · Simplification functions ·

Function: trigrat (expr) ¶

Gives a canonical simplified 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$

(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                           %pi
                 sin(a) sin(3 ((- b) - a + ---))
                                            3
(%o2)            -------------------------------
                           sin(b + a)

(%i3) ba : bc, c=a, a=c;
                                         %pi
                    sin(3 a) sin(b + a - ---)
                                          3
(%o3)               -------------------------
                                  %pi
                          sin(a - ---)
                                   3

(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
         2         2         %pi
      sin (3 a) sin (b + a - ---)
                              3
(%o4) ---------------------------
                2     %pi
             sin (a - ---)
                       3
                                         %pi
 - (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
                                          3
             %pi            %pi
 sin(b + a - ---))/(sin(a - ---) sin(b + a))
              3              3
      2       2                %pi
   sin (a) sin (3 ((- b) - a + ---))
                                3
 + ---------------------------------
                 2
              sin (b + a)

(%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

Categories: Trigonometric functions · Simplification functions ·