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]