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The Maxima on-line user's manual

Algebra Calculator

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Halfangles Calculator

Halfangles

-- Option variable: halfangles Default value: false

When halfangles is true, trigonometric functions of arguments <expr>/2 are simplified to functions of <expr>.

For a real argument <x> in the interval 0 < x < 2*%pi the sine of the half-angle simplifies to a simple formula:

sqrt(1 - cos(x)) ---------------- sqrt(2)

A complicated factor is needed to make this formula correct for all complex arguments <z>:

realpart(z) floor(-----------) 2 %pi (- 1) (1 - unit_step(- imagpart(z))

realpart(z) realpart(z) floor(-----------) - ceiling(-----------) 2 %pi 2 %pi ((- 1) + 1))

Maxima knows this factor and similar factors for the functions sin, cos, sinh, and cosh. For special values of the argument z these factors simplify accordingly.

Examples:

          (%i1) halfangles:false;
          (%o1)                                false
          (%i2) sin(x/2);
                                                  x
          (%o2)                               sin(-)
                                                  2
          (%i3) halfangles:true;
          (%o3)                                true
          (%i4) sin(x/2);
                                                              x
                                                      floor(-----)
                                                            2 %pi
                                sqrt(1 - cos(x)) (- 1)
          (%o4)                 ----------------------------------
                                             sqrt(2)
          (%i5) assume(x>0, x<2*%pi)$
          (%i6) sin(x/2);
                                         sqrt(1 - cos(x))
          (%o6)                          ----------------
                                             sqrt(2)

(%o1)                                true
(%i2) 

Halfangles Example

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halfangles

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halfangles

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