trigtools
atan_contract (r) — Function
The function atan_contract(r) contracts atan functions. We assume: $|r| < {\pi\over 2}.$
load("trigtools") loads this function.
Examples:
maxima (%i1) load(“trigtools”)$
(%i2) atan_contract(atan(x)+atan(y)); (%o2) atan(y) + atan(x)
(%i3) assume(abs(atan(x)+atan(y))<%pi/2)$
(%i4) atan(x)+atan(y)=atan_contract(atan(x)+atan(y)); y + x (%o4) atan(y) + atan(x) = atan(—––) 1 - x y
2. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) atan(1/3)+atan(1/5)+atan(1/7)+atan(1/8)$ %=atan_contract(%);
1 1 1 1 %pi
(%o3) atan(-) + atan(-) + atan(-) + atan(-) = ---
3 5 7 8 4
- Machin’s formulae
maxima
(%i1) load("trigtools")$
(%i2) 4*atan(1/5)-atan(1/239)=atan_contract(4*atan(1/5)-atan(1/239));
1 1 %pi
(%o2) 4 atan(-) - atan(---) = ---
5 239 4
maxima
(%i1) load("trigtools")$
(%i2) 12*atan(1/49)+32*atan(1/57)-5*atan(1/239)+12*atan(1/110443)$
(%i3) %=atan_contract(%);
1 1 1
(%o3) 12 atan(--) + 32 atan(--) - 5 atan(---)
49 57 239
1 %pi
+ 12 atan(------) = ---
110443 4
c2hyp (x) — Function
The function c2hyp (convert to hyperbolic) convert expression with exp function to expression with hyperbolic functions sinh, cosh.
load("trigtools") loads this function.
Examples:
maxima
(%i1) load("trigtools")$
(%i2) c2hyp(exp(x));
(%o2) sinh(x) + cosh(x)
(%i3) c2hyp(exp(x)+exp(x^2)+1);
2 2
(%o3) sinh(x ) + cosh(x ) + sinh(x) + cosh(x) + 1
(%i4) c2hyp(exp(x)/(2*exp(y)-3*exp(z)));
sinh(x) + cosh(x)
(%o4) ---------------------------------------------
2 (sinh(y) + cosh(y)) - 3 (sinh(z) + cosh(z))
c2sin (x) — Function
The function c2sin converts the expression $a\cos x + b\sin x$ to $r\sin(x+\phi).$
The function c2cos converts the expression $a\cos x + b\sin x$ to $r\cos(x-\phi).$
load("trigtools") loads these functions.
Examples:
maxima
(%i1) load("trigtools")$
(%i2) c2sin(3*sin(x)+4*cos(x));
4
(%o2) 5 sin(x + atan(-))
3
(%i3) trigexpand(%),expand;
(%o3) 3 sin(x) + 4 cos(x)
(%i4) c2cos(3*sin(x)-4*cos(x));
3
(%o4) - 5 cos(x + atan(-))
4
(%i5) trigexpand(%),expand;
(%o5) 3 sin(x) - 4 cos(x)
(%i6) c2sin(sin(x)+cos(x));
%pi
(%o6) sqrt(2) sin(x + ---)
4
(%i7) trigexpand(%),expand;
(%o7) sin(x) + cos(x)
(%i8) c2cos(sin(x)+cos(x));
%pi
(%o8) sqrt(2) cos(x - ---)
4
(%i9) trigexpand(%),expand;
(%o9) sin(x) + cos(x)
Example. Solve trigonometric equation
maxima
(%i1) eq:3*sin(x)+4*cos(x)=2;
(%o1) 3 sin(x) + 4 cos(x) = 2
(%i2) plot2d([3*sin(x)+4*cos(x),2],[x,-%pi,%pi]);
(%o2) false
maxima
(%i1) load("trigtools")$
(%i2) eq:3*sin(x)+4*cos(x)=2$
(%i3) eq1:c2sin(lhs(eq))=2;
4
(%o3) 5 sin(x + atan(-)) = 2
3
(%i4) solvetrigwarn:false$
(%i5) solve(eq1)[1]$ x1:rhs(%);
2 4
(%o6) asin(-) - atan(-)
5 3
(%i7) float(%), numer;
(%o7) - 0.5157783719341241
(%i8) eq2:c2cos(lhs(eq))=2;
3
(%o8) 5 cos(x - atan(-)) = 2
4
(%i9) solve(eq2,x)[1]$ x2:rhs(%);
3 2
(%o10) atan(-) + acos(-)
4 5
(%i11) float(%), numer;
(%o11) 1.802780589520693
(%i12) sol:[x1,x2];
2 4 3 2
(%o12) [asin(-) - atan(-), atan(-) + acos(-)]
5 3 4 5
Answ.: $x = x_1 + 2\pi k,$ $x_1 = \sin^{-1}{2\over 5} - \tan^{-1}{4\over 3},$ or $x_1 = \tan^{-1}{3\over 4} + \cos^{-1}{2\over 5},$ for $k$ any integer.
c2trig (x) — Function
The function c2trig (convert to trigonometric) reduce expression with hyperbolic functions sinh, cosh, tanh, coth to trigonometric expression with sin, cos, tan, cot.
load("trigtools") loads these functions.
Examples:
maxima (%i1) load(trigtools)$
(%i2) sinh(x)=c2trig(sinh(x)); (%o2) sinh(x) = - %i sin(%i x)
(%i3) cosh(x)=c2trig(cosh(x)); (%o3) cosh(x) = cos(%i x)
(%i4) tanh(x)=c2trig(tanh(x)); (%o4) tanh(x) = - %i tan(%i x)
(%i5) coth(x)=c2trig(coth(x)); (%o5) coth(x) = %i cot(%i x)
2. see [https://maxima.sourceforge.io/ext/list_archives/2013/msg03230.html]()
```maxima
maxima
(%i1) load("trigtools")$
(%i2) cos(p+q*%i);
(%o2) cos(%i q + p)
(%i3) trigexpand(%);
(%o3) cos(p) cosh(q) - %i sin(p) sinh(q)
(%i4) c2trig(%);
(%o4) cos(%i q + p)
maxima (%i1) load(“trigtools”)$
(%i2) sin(a+b*%i); (%o2) sin(%i b + a)
(%i3) trigexpand(%); (%o3) %i cos(a) sinh(b) + sin(a) cosh(b)
(%i4) c2trig(%); (%o4) sin(%i b + a)
4. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) cos(a*%i+b*%i);
(%o2) cos(%i b + %i a)
(%i3) trigexpand(%);
(%o3) sinh(a) sinh(b) + cosh(a) cosh(b)
(%i4) c2trig(%);
(%o4) cos(%i b + %i a)
maxima (%i1) load(“trigtools”)$
(%i2) tan(a+%i*b); (%o2) tan(%i b + a)
(%i3) trigexpand(%); %i tanh(b) + tan(a) (%o3) ——————— 1 - %i tan(a) tanh(b)
(%i4) c2trig(%); (%o4) tan(%i b + a)
6. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) cot(x+%i*y);
(%o2) cot(%i y + x)
(%i3) trigexpand(%);
- %i cot(x) coth(y) - 1
(%o3) -----------------------
cot(x) - %i coth(y)
(%i4) c2trig(%);
(%o4) cot(%i y + x)
trigeval (x) — Function
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.
trigfactor (x) — Function
The function trigfactor factors expressions of form $\pm \sin x \pm \cos y.$
load("trigtools") loads this function.
Examples:
maxima (%i1) load(“trigtools”)$
(%i2) trigfactor(sin(x)+cos(x)); %pi (%o2) sqrt(2) cos(x - —) 4
(%i3) trigrat(%); (%o3) sin(x) + cos(x)
2. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) trigfactor(sin(x)+cos(y));
y x %pi y x %pi
(%o2) 2 cos(- - - + ---) cos(- + - - ---)
2 2 4 2 2 4
(%i3) trigrat(%);
(%o3) cos(y) + sin(x)
maxima (%i1) load(“trigtools”)$
(%i2) trigfactor(sin(x)-cos(3*y)); 3 y x %pi 3 y x %pi (%o2) 2 sin(— - - + —) sin(— + - - —) 2 2 4 2 2 4
(%i3) trigrat(%); (%o3) sin(x) - cos(3 y)
4. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) trigfactor(-sin(5*x)-cos(3*y));
3 y 5 x %pi 3 y 5 x %pi
(%o2) - 2 cos(--- - --- + ---) cos(--- + --- - ---)
2 2 4 2 2 4
(%i3) trigrat(%);
(%o3) - cos(3 y) - sin(5 x)
maxima (%i1) load(“trigtools”)$
(%i2) sin(alpha)+sin(beta)=trigfactor(sin(alpha)+sin(beta)); beta alpha (%o2) sin(beta) + sin(alpha) = 2 cos(–– - —–) 2 2 beta alpha sin(–– + —–) 2 2
(%i3) trigrat(%); (%o3) sin(beta) + sin(alpha) = sin(beta) + sin(alpha)
6. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) sin(alpha)-sin(beta)=trigfactor(sin(alpha)-sin(beta));
beta alpha
(%o2) sin(alpha) - sin(beta) = - 2 sin(---- - -----)
2 2
beta alpha
cos(---- + -----)
2 2
maxima (%i1) load(“trigtools”)$
(%i2) cos(alpha)+cos(beta)=trigfactor(cos(alpha)+cos(beta)); beta alpha (%o2) cos(beta) + cos(alpha) = 2 cos(–– - —–) 2 2 beta alpha cos(–– + —–) 2 2
8. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) cos(alpha)-cos(beta)=trigfactor(cos(alpha)-cos(beta));
beta alpha
(%o2) cos(alpha) - cos(beta) = 2 sin(---- - -----)
2 2
beta alpha
sin(---- + -----)
2 2
maxima (%i1) load(“trigtools”)$
(%i2) trigfactor(3sin(x)+7cos(x)); (%o2) 3 sin(x) + 7 cos(x)
(%i3) c2sin(%); 7 (%o3) sqrt(58) sin(x + atan(-)) 3
(%i4) trigexpand(%),expand; (%o4) 3 sin(x) + 7 cos(x)
10. ```maxima
maxima
(%i1) load("trigtools")$
(%i2) trigfactor(sin(2*x));
(%o2) sin(2 x)
(%i3) trigexpand(%);
(%o3) 2 cos(x) sin(x)
trigsolve (x) — Function
The function trigsolve find solutions of trigonometric equation from interval $[a,b).$
load("trigtools") loads this function.
Examples:
maxima
(%i1) eq:eq:3sin(x)+4cos(x)=2; (%o1) 3 sin(x) + 4 cos(x) = 2
(%i2) plot2d([3sin(x)+4cos(x),2],[x,-%pi,%pi]); (%o2) false
```maxima
maxima
(%i1) load("trigtools")$
(%i2) eq:eq:3*sin(x)+4*cos(x)=2$
(%i3) sol:trigsolve(eq,-%pi,%pi);
2 sqrt(21) 12 2 sqrt(21) 12
(%o3) {atan(---------- - --), %pi - atan(---------- + --)}
5 5 5 5
(%i4) float(%), numer;
(%o4) {- 0.5157783719341241, 1.8027805895206928}
Answ. : $x = \tan^{-1}\left({2\sqrt{21}\over 5} - {12\over 5}\right) + 2\pi k$ ; $x = \pi - \tan^{-1}\left({2\sqrt{21}\over 5} + {12\over 5}\right) + 2\pi k,$ $k$ – any integer. 2. ```maxima maxima (%i1) load(“trigtools”)$
(%i2) eq:cos(3x)-sin(x)=sqrt(3)(cos(x)-sin(3*x)); (%o2) cos(3 x) - sin(x) = sqrt(3) (cos(x) - sin(3 x))
(%i3) plot2d([lhs(eq)-rhs(eq)], [x,0,2*%pi])$
We have 6 solutions from [0, 2*pi].
```maxima
maxima
(%i1) load("trigtools")$
(%i2) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x))$
(%i3) plot2d([lhs(eq)-rhs(eq)], [x,0.2,0.5])$
maxima
(%i1) load("trigtools")$
(%i2) load("trigtools")$
(%i3) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x))$
(%i4) plot2d([lhs(eq)-rhs(eq)], [x,3.3,3.6])$
maxima
(%i1) load("trigtools")$
(%i2) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x))$
(%i3) trigfactor(lhs(eq))=map(trigfactor,rhs(eq));
%pi %pi
(%o3) - 2 sin(x + ---) sin(2 x - ---) =
4 4
%pi %pi
2 sqrt(3) sin(x - ---) sin(2 x - ---)
4 4
(%i4) factor(lhs(%)-rhs(%));
4 x + %pi 4 x - %pi
(%o4) - 2 (sin(---------) + sqrt(3) sin(---------))
4 4
8 x - %pi
sin(---------)
4
Equation is equivalent to
maxima
(%i1) load("trigtools")$
(%i2) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x))$
(%i3) trigfactor(lhs(eq))=map(trigfactor,rhs(eq));
%pi %pi
(%o3) - 2 sin(x + ---) sin(2 x - ---) =
4 4
%pi %pi
2 sqrt(3) sin(x - ---) sin(2 x - ---)
4 4
(%i4) L:factor(rhs(%)-lhs(%));
4 x + %pi 4 x - %pi 8 x - %pi
(%o4) 2 (sin(---------) + sqrt(3) sin(---------)) sin(---------)
4 4 4
(%i5) eq1:part(L,2)=0;
4 x + %pi 4 x - %pi
(%o5) sin(---------) + sqrt(3) sin(---------) = 0
4 4
(%i6) eq2:part(L,3)=0;
8 x - %pi
(%o6) sin(---------) = 0
4
(%i7) S1:trigsolve(eq1,0,2*%pi);
%pi 13 %pi
(%o7) {---, ------}
12 12
(%i8) S2:trigsolve(eq2,0,2*%pi);
%pi 5 %pi 9 %pi 13 %pi
(%o8) {---, -----, -----, ------}
8 8 8 8
(%i9) S:listify(union(S1,S2));
%pi %pi 5 %pi 13 %pi 9 %pi 13 %pi
(%o9) [---, ---, -----, ------, -----, ------]
12 8 8 12 8 8
(%i10) float(%), numer;
(%o10) [0.2617993877991494, 0.39269908169872414,
1.9634954084936207, 3.4033920413889422, 3.5342917352885173,
5.105088062083414]
Answer: $x = a + 2\pi k,$ where $a$ any from $S$, $k$ any integer. 3. ```maxima maxima (%i1) load(“trigtools”)$
(%i2) eq:8*cos(x)cos(4x)cos(5x)-1=0; (%o2) 8 cos(x) cos(4 x) cos(5 x) - 1 = 0
(%i3) trigrat(%); (%o3) 2 cos(10 x) + 2 cos(8 x) + 2 cos(2 x) + 1 = 0
Left side is periodic with period
$T=\pi.$
We have 10 solutions from [0, pi].
```maxima
maxima
(%i1) load("trigtools")$
(%i2) eq:8*cos(x)*cos(4*x)*cos(5*x)-1=0$
(%i3) plot2d([lhs(eq),rhs(eq)],[x,0,%pi]);
(%o3) false
maxima
(%i1) load("trigtools")$
(%i2) eq:8*cos(x)*cos(4*x)*cos(5*x)-1=0$
(%i3) x4:find_root(eq, x, 1.3, 1.32);
(%o3) 1.3089969389957472
(%i4) x5:find_root(eq, x, 1.32, 1.35);
(%o4) 1.3463968515384828
(%i5) plot2d([lhs(eq),0], [x,1.3,1.35], [gnuplot_preamble, "set grid;"]);
(%o5) false
Equation we multiply by $2\sin x\cos 2x:$
maxima
(%i1) load("trigtools")$
(%i2) eq:8*cos(x)*cos(4*x)*cos(5*x)-1=0$
(%i3) eq*2*sin(x)*cos(2*x);
(%o3) 2 sin(x) cos(2 x) (8 cos(x) cos(4 x) cos(5 x) - 1) = 0
(%i4) eq1:trigreduce(%),expand;
(%o4) sin(13 x) + sin(x) = 0
(%i5) trigfactor(lhs(eq1))=0;
(%o5) 2 cos(6 x) sin(7 x) = 0
(%i6) S1:trigsolve(cos(6*x),0,%pi);
%pi %pi 5 %pi 7 %pi 3 %pi 11 %pi
(%o6) {---, ---, -----, -----, -----, ------}
12 4 12 12 4 12
(%i7) S2:trigsolve(sin(7*x),0,%pi);
%pi 2 %pi 3 %pi 4 %pi 5 %pi 6 %pi
(%o7) {0, ---, -----, -----, -----, -----, -----}
7 7 7 7 7 7
We remove solutions of $\sin x = 0$ and $\cos 2x = 0.$
maxima
(%i1) load("trigtools")$
(%i2) S1:trigsolve(cos(6*x),0,%pi)$
(%i3) S2:trigsolve(sin(7*x),0,%pi)$
(%i4) S3:trigsolve(sin(x),0,%pi);
(%o4) {0}
(%i5) S4:trigsolve(cos(2*x),0,%pi);
%pi 3 %pi
(%o5) {---, -----}
4 4
We find 10 solutions from $[0, \pi]:$
maxima
(%i1) load("trigtools")$
(%i2) S1:trigsolve(cos(6*x),0,%pi)$
(%i3) S2:trigsolve(sin(7*x),0,%pi)$
(%i4) S3:trigsolve(sin(x),0,%pi)$
(%i5) S4:trigsolve(cos(2*x),0,%pi)$
(%i6) union(S1,S2)$ setdifference(%,S3)$ setdifference(%,S4);
%pi %pi 2 %pi 5 %pi 3 %pi 4 %pi 7 %pi 5 %pi
(%o8) {---, ---, -----, -----, -----, -----, -----, -----,
12 7 7 12 7 7 12 7
6 %pi 11 %pi
-----, ------}
7 12
(%i9) S:listify(%);
%pi %pi 2 %pi 5 %pi 3 %pi 4 %pi 7 %pi 5 %pi
(%o9) [---, ---, -----, -----, -----, -----, -----, -----,
12 7 7 12 7 7 12 7
6 %pi 11 %pi
-----, ------]
7 12
(%i10) length(S);
(%o10) 10
(%i11) float(S), numer;
(%o11) [0.2617993877991494, 0.4487989505128276,
0.8975979010256552, 1.3089969389957472, 1.3463968515384828,
1.7951958020513104, 1.8325957145940461, 2.243994752564138,
2.6927937030769655, 2.8797932657906435]
Answer: $x = a + 2\pi k,$ where $a$ any from $S$, $k$ any integer.
trigvalue (x) — Function
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.
trigvalue is essentially an internal function.
Use function trigeval in preference.
load("trigtools") loads this function.
See also: trigeval.