96.2.6 Evaluation of Trignometric Functions

Function: trigvalue (x) ¶

The function trigvalue compute values of \(\sin {m\pi\over n},\) \(\cos {m\pi\over n},\) \(\tan {m\pi\over n},\) and \(\cot {m\pi\over n}\) in radicals.

Function: trigeval (x) ¶

The function trigeval compute values of expressions with \(\sin {m\pi\over n},\) \(\cos {m\pi\over n},\) \(\tan {m\pi\over n},\) and \(\cot {m\pi\over n}\) in radicals.

Examples:

1. Values of trignometric functions

   (%i1) load(trigtools)$
   

   (%i2) trigvalue(sin(%pi/10));
                                     sqrt(5) - 1
   (%o2)                             -----------
                                          4
   

   (%i3) trigvalue(cos(%pi/10));
                                  sqrt(sqrt(5) + 5)
   (%o3)                          -----------------
                                         3/2
                                        2
   

   (%i4) trigvalue(tan(%pi/10));
                                 sqrt(5 - 2 sqrt(5))
   (%o4)                         -------------------
                                       sqrt(5)
   

   (%i5) float(%), numer;
   (%o5)                         0.3249196962329063
   (%i6) float(tan(%pi/10)), numer;
   (%o6)                         0.3249196962329063
   (%i7) trigvalue(cot(%pi/10));
   (%o7)                         sqrt(2 sqrt(5) + 5)
   (%i8) float(%), numer;
   (%o8)                          3.077683537175254
   (%i9) float(cot(%pi/10)), numer;
   (%o9)                          3.077683537175254
   (%i10) trigvalue(sin(%pi/32));
                        sqrt(2 - sqrt(sqrt(sqrt(2) + 2) + 2))
   (%o10)               -------------------------------------
                                          2
   (%i11) trigvalue(cos(%pi/32));
                        sqrt(sqrt(sqrt(sqrt(2) + 2) + 2) + 2)
   (%o11)               -------------------------------------
                                          2
   (%i12) trigvalue(cos(%pi/256));
          sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2) + 2) + 2) + 2) + 2) + 2) + 2)
   (%o12) -------------------------------------------------------------------
                                           2
   (%i13) trigvalue(cos(%pi/60));
           sqrt(sqrt(sqrt(2) sqrt(3) sqrt(sqrt(5) + 5) + sqrt(5) + 7) + 4)
   (%o13)  ---------------------------------------------------------------
                                         3/2
                                        2
   

   (%i14) trigvalue(sin(%pi/60));
           sqrt(4 - sqrt(sqrt(2) sqrt(3) sqrt(sqrt(5) + 5) + sqrt(5) + 7))
   (%o14)  ---------------------------------------------------------------
                                         3/2
                                        2
   

   (%i15) trigvalue(sin(%pi/18));
                                          %pi
   (%o15)                             sin(---)
                                          18
   

   (%i16) trigvalue(sin(%pi/20));
                         sqrt(4 - sqrt(2) sqrt(sqrt(5) + 5))
   (%o16)                -----------------------------------
                                         3/2
                                        2
   

2. ode example

   (%i17) load(odes)$
   

   (%i18) eq:'diff(y,x,5)+2*y=0;
                                     5
                                    d y
   (%o18)                           --- + 2 y = 0
                                      5
                                    dx
   

   (%i19) odeL(eq,y,x);
   

                      1/5     4 %pi
                   - 2    cos(-----) x
                                5           1/5     4 %pi
   (%o19) y = C5 %e                    sin(2    sin(-----) x)
                                                      5
              1/5     4 %pi
           - 2    cos(-----) x
                        5           1/5     4 %pi
    + C4 %e                    cos(2    sin(-----) x)
                                              5
              1/5     2 %pi
           - 2    cos(-----) x
                        5           1/5     2 %pi
    + C3 %e                    sin(2    sin(-----) x)
                                              5
              1/5     2 %pi
           - 2    cos(-----) x                                  1/5
                        5           1/5     2 %pi            - 2    x
    + C2 %e                    cos(2    sin(-----) x) + C1 %e
                                              5
   

   (%i20) sol:trigeval(%);
                     (sqrt(5) - 1) x
                   - ---------------
                           9/5
                          2              sqrt(sqrt(5) + 5) x
   (%o20) y = C3 %e                  sin(-------------------)
                                                13/10
                                               2
             (sqrt(5) - 1) x
           - ---------------
                   9/5
                  2              sqrt(sqrt(5) + 5) x
    + C2 %e                  cos(-------------------)
                                        13/10
                                       2
           (sqrt(5) + 1) x
           ---------------
                 9/5
                2              sqrt(5 - sqrt(5)) x
    + C5 %e                sin(-------------------)
                                      13/10
                                     2
           (sqrt(5) + 1) x
           ---------------
                 9/5                                          1/5
                2              sqrt(5 - sqrt(5)) x         - 2    x
    + C4 %e                cos(-------------------) + C1 %e
                                      13/10
                                     2
   

   (%i21) subst(sol,eq)$
   (%i22) ev(%, nouns)$
   (%i23) radcan(%);
   (%o23)                               0 = 0
   

3. n-th root of complex number

   Example. Find the 4-th roots of %i

   (%i24) solve(x^4=%i,x);
                    1/8                1/8             1/8              1/8
   (%o24) [x = (- 1)    %i, x = - (- 1)   , x = - (- 1)    %i, x = (- 1)   ]
   

   (%i25) rectform(%);
                      %pi        %pi                 %pi         %pi
   (%o25) [x = %i cos(---) - sin(---), x = (- %i sin(---)) - cos(---), 
                       8          8                   8           8
                                   %pi           %pi              %pi        %pi
                           x = sin(---) - %i cos(---), x = %i sin(---) + cos(---)]
                                    8             8                8          8
   

   (%i26) trigeval(%);
               sqrt(sqrt(2) + 2) %i   sqrt(2 - sqrt(2))
   (%o26) [x = -------------------- - -----------------, 
                        2                     2
          sqrt(2 - sqrt(2)) %i    sqrt(sqrt(2) + 2)
   x = (- --------------------) - -----------------, 
                   2                      2
       sqrt(2 - sqrt(2))   sqrt(sqrt(2) + 2) %i
   x = ----------------- - --------------------, 
               2                    2
       sqrt(2 - sqrt(2)) %i   sqrt(sqrt(2) + 2)
   x = -------------------- + -----------------]
                2                     2