Next: Contract atan Functions, Previous: Solve Trignometric Equations, Up: Functions and Variables for trigtools [Contents][Index]
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.
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:
(%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
(%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
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
Next: Contract atan Functions, Previous: Solve Trignometric Equations, Up: Functions and Variables for trigtools [Contents][Index]