99 Package trigtools ¶
Trigtools Package
Aleksas Dormarkas
December 1, 2013
99.1 Introduction to trigtools ¶
We use open-source computer algebra system(CAS) maxima 5.31.2. The trigtools package10 contains commands that help you work with trigonometric expressions. List of functions in trigtools package:
Categories:
Trigonometric functions
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Package trigtools
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99.2 Functions and Variables for trigtools ¶
- Convert to sin and cos
- Convert to Trigonometric Functions
- Convert to Hyperbolic Functions
- Factor Sums of sin and cos Functions
- Solve Trigonometric Equations
- Evaluation of Trigonometric Functions
- Contract atan Functions
99.2.1 Convert to sin and cos ¶
- Function: c2sin (x) ¶
- Function: c2cos (x) ¶
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:
(%i1) load("trigtools")$(%i2) c2sin(3*sin(x)+4*cos(x)); 4 (%o2) 5 sin(x + atan(-)) 3(%i3) trigexpand(%),expand; (%o3) 3 sin(x) + 4 cos(x)
(%i4) c2cos(3*sin(x)-4*cos(x)); 3 (%o4) - 5 cos(x + atan(-)) 4(%i5) trigexpand(%),expand; (%o5) 3 sin(x) - 4 cos(x)
(%i6) c2sin(sin(x)+cos(x)); %pi (%o6) sqrt(2) sin(x + ---) 4(%i7) trigexpand(%),expand; (%o7) sin(x) + cos(x)
(%i8) c2cos(sin(x)+cos(x)); %pi (%o8) sqrt(2) cos(x - ---) 4(%i9) trigexpand(%),expand; (%o9) sin(x) + cos(x)
Example. Solve trigonometric equation
(%i1) eq:3*sin(x)+4*cos(x)=2; (%o1) 3 sin(x) + 4 cos(x) = 2
(%i2) plot2d([3*sin(x)+4*cos(x),2],[x,-%pi,%pi]); (%o2) false
(%i1) load("trigtools")$ (%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 5Answ.: \(x = x_1 + 2\pi k,\) \(x_1 = \sin^{-1}{2\over 5} - \tan^{-1}{4\over 3},\) or \(x_1 = \tan^{-1}{3\over 4} + \cos^{-1}{2\over 5},\) for \(k\) any integer.
99.2.2 Convert to Trigonometric Functions ¶
- Function: c2trig (x) ¶
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:
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(%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)
- see https://maxima.sourceforge.io/ext/list_archives/2013/msg03230.html
(%i1) load("trigtools")$(%i2) cos(p+q*%i); (%o2) cos(%i q + p)
(%i3) trigexpand(%); (%o3) cos(p) cosh(q) - %i sin(p) sinh(q)
(%i4) c2trig(%); (%o4) cos(%i q + p)
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(%i1) load("trigtools")$(%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)
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(%i1) load("trigtools")$(%i2) cos(a*%i+b*%i); (%o2) cos(%i b + %i a)
(%i3) trigexpand(%); (%o3) sinh(a) sinh(b) + cosh(a) cosh(b)
(%i4) c2trig(%); (%o4) cos(%i b + %i a)
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(%i1) load("trigtools")$(%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)
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(%i1) load("trigtools")$(%i2) cot(x+%i*y); (%o2) cot(%i y + x)
(%i3) trigexpand(%); - %i cot(x) coth(y) - 1 (%o3) ----------------------- cot(x) - %i coth(y)(%i4) c2trig(%); (%o4) cot(%i y + x)
Categories: Trigonometric functions · Package trigtools ·-
99.2.3 Convert to Hyperbolic Functions ¶
- Function: c2hyp (x) ¶
The function c2hyp (convert to hyperbolic) convert expression with exp function to expression with hyperbolic functions sinh, cosh.
load("trigtools")loads this function.Examples:
(%i1) load("trigtools")$(%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))
99.2.4 Factor Sums of sin and cos Functions ¶
- Function: trigfactor (x) ¶
The function trigfactor factors expressions of form \(\pm \sin x \pm \cos y.\)
load("trigtools")loads this function.Examples:
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(%i1) load("trigtools")$(%i2) trigfactor(sin(x)+cos(x)); %pi (%o2) sqrt(2) cos(x - ---) 4(%i3) trigrat(%); (%o3) sin(x) + cos(x)
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(%i1) load("trigtools")$(%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)
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(%i1) load("trigtools")$(%i2) trigfactor(sin(x)-cos(3*y)); 3 y x %pi 3 y x %pi (%o2) 2 sin(--- - - + ---) sin(--- + - - ---) 2 2 4 2 2 4(%i3) trigrat(%); (%o3) sin(x) - cos(3 y)
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(%i1) load("trigtools")$(%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)
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(%i1) load("trigtools")$(%i2) sin(alpha)+sin(beta)=trigfactor(sin(alpha)+sin(beta)); beta alpha (%o2) sin(beta) + sin(alpha) = 2 cos(---- - -----) 2 2 beta alpha sin(---- + -----) 2 2(%i3) trigrat(%); (%o3) sin(beta) + sin(alpha) = sin(beta) + sin(alpha)
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(%i1) load("trigtools")$(%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 -
(%i1) load("trigtools")$(%i2) cos(alpha)+cos(beta)=trigfactor(cos(alpha)+cos(beta)); beta alpha (%o2) cos(beta) + cos(alpha) = 2 cos(---- - -----) 2 2 beta alpha cos(---- + -----) 2 2 -
(%i1) load("trigtools")$(%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 -
(%i1) load("trigtools")$(%i2) trigfactor(3*sin(x)+7*cos(x)); (%o2) 3 sin(x) + 7 cos(x)
(%i3) c2sin(%); 7 (%o3) sqrt(58) sin(x + atan(-)) 3(%i4) trigexpand(%),expand; (%o4) 3 sin(x) + 7 cos(x)
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(%i1) load("trigtools")$(%i2) trigfactor(sin(2*x)); (%o2) sin(2 x)
(%i3) trigexpand(%); (%o3) 2 cos(x) sin(x)
Categories: Trigonometric functions · Package trigtools ·-
99.2.5 Solve Trigonometric Equations ¶
- Function: trigsolve (x) ¶
The function trigsolve find solutions of trigonometric equation from interval \([a,b).\)
load("trigtools")loads this function.Examples:
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(%i1) eq: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
(%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.
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(%i1) load("trigtools")$(%i2) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x)); (%o2) cos(3 x) - sin(x) = sqrt(3) (cos(x) - sin(3 x))
(%i3) plot2d([lhs(eq)-rhs(eq)], [x,0,2*%pi])$
We have 6 solutions from [0, 2*pi].
(%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])$
(%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])$
(%i1) load("trigtools")$ (%i2) eq:cos(3*x)-sin(x)=sqrt(3)*(cos(x)-sin(3*x))$(%i3) trigfactor(lhs(eq))=map(trigfactor,rhs(eq)); %pi %pi (%o3) - 2 sin(x + ---) sin(2 x - ---) = 4 4 %pi %pi 2 sqrt(3) sin(x - ---) sin(2 x - ---) 4 4(%i4) factor(lhs(%)-rhs(%)); 4 x + %pi 4 x - %pi (%o4) - 2 (sin(---------) + sqrt(3) sin(---------)) 4 4 8 x - %pi sin(---------) 4Equation is equivalent to
(%i1) load("trigtools")$ (%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.
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(%i1) load("trigtools")$(%i2) eq:8*cos(x)*cos(4*x)*cos(5*x)-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].
(%i1) load("trigtools")$ (%i2) eq:8*cos(x)*cos(4*x)*cos(5*x)-1=0$(%i3) plot2d([lhs(eq),rhs(eq)],[x,0,%pi]); (%o3) false
(%i1) load("trigtools")$ (%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 \(2\sin x\cos 2x:\)
(%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 7We remove solutions of \(\sin x = 0\) and \(\cos 2x = 0.\)
(%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 4We find 10 solutions from \([0, \pi]:\)
(%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.
Categories: Trigonometric functions · Package trigtools ·-
99.2.6 Evaluation of Trigonometric 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.
trigvalueis essentially an internal function. Use functiontrigevalin preference.load("trigtools")loads this function.
- 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.
load("trigtools") loads this function.
Examples:
- Values of trigonometric functions
(%i1) load("trigtools")$ (%i2) load(trigtools)$(%i3) trigvalue(sin(%pi/10)); sqrt(5) - 1 (%o3) ----------- 4(%i4) trigvalue(cos(%pi/10)); sqrt(sqrt(5) + 5) (%o4) ----------------- 3/2 2(%i5) trigvalue(tan(%pi/10)); sqrt(5 - 2 sqrt(5)) (%o5) ------------------- sqrt(5)(%i6) float(%), numer; (%o6) 0.3249196962329063
(%i7) float(tan(%pi/10)), numer; (%o7) 0.3249196962329063
(%i8) trigvalue(cot(%pi/10)); (%o8) sqrt(2 sqrt(5) + 5)
(%i9) float(%), numer; (%o9) 3.0776835371752536
(%i10) float(cot(%pi/10)), numer; (%o10) 3.077683537175254
(%i11) trigvalue(sin(%pi/32)); sqrt(2 - sqrt(sqrt(sqrt(2) + 2) + 2)) (%o11) ------------------------------------- 2(%i12) trigvalue(cos(%pi/32)); sqrt(sqrt(sqrt(sqrt(2) + 2) + 2) + 2) (%o12) ------------------------------------- 2(%i13) trigvalue(cos(%pi/256)); (%o13) sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2) + 2) + 2) + 2) + 2) + 2) + 2)/2(%i14) trigvalue(cos(%pi/60)); (%o14) sqrt(sqrt(sqrt(2) sqrt(3) sqrt(sqrt(5) + 5) + sqrt(5) + 7) + 4) --------------------------------------------------------------- 3/2 2(%i15) trigvalue(sin(%pi/60)); (%o15) sqrt(4 - sqrt(sqrt(2) sqrt(3) sqrt(sqrt(5) + 5) + sqrt(5) + 7)) --------------------------------------------------------------- 3/2 2(%i16) trigvalue(sin(%pi/18)); %pi (%o16) sin(---) 18(%i17) trigvalue(sin(%pi/20)); sqrt(4 - sqrt(2) sqrt(sqrt(5) + 5)) (%o17) ----------------------------------- 3/2 2 - ode example
(%i1) load("trigtools")$ (%i2) load(odes)$(%i3) eq:'diff(y,x,5)+2*y=0; 5 d y (%o3) --- + 2 y = 0 5 dx(%i4) odeL(eq,y,x); 1/5 4 %pi - 2 cos(-----) x 5 1/5 4 %pi (%o4) y = %e C5 sin(2 sin(-----) x) 5 1/5 4 %pi - 2 cos(-----) x 5 1/5 4 %pi + %e C4 cos(2 sin(-----) x) 5 1/5 2 %pi - 2 cos(-----) x 5 1/5 2 %pi + %e C3 sin(2 sin(-----) x) 5 1/5 2 %pi - 2 cos(-----) x 5 1/5 2 %pi + %e C2 cos(2 sin(-----) x) 5 1/5 - 2 x + %e C1(%i5) sol:trigeval(%); (sqrt(5) - 1) x - --------------- 9/5 2 sqrt(sqrt(5) + 5) x (%o5) y = %e C3 sin(-------------------) 13/10 2 (sqrt(5) - 1) x - --------------- 9/5 2 sqrt(sqrt(5) + 5) x + %e C2 cos(-------------------) 13/10 2 (sqrt(5) + 1) x --------------- 9/5 2 sqrt(5 - sqrt(5)) x + %e C5 sin(-------------------) 13/10 2 (sqrt(5) + 1) x --------------- 9/5 1/5 2 sqrt(5 - sqrt(5)) x - 2 x + %e C4 cos(-------------------) + %e C1 13/10 2(%i6) subst(sol,eq)$ (%i7) ev(%, nouns)$
(%i8) radcan(%); (%o8) 0 = 0
- n-th root of complex number
Example. Find the 4-th roots of %i
(%i1) load("trigtools")$(%i2) solve(x^4=%i,x); 1/8 1/8 1/8 (%o2) [x = (- 1) %i, x = - (- 1) , x = - (- 1) %i, 1/8 x = (- 1) ](%i3) rectform(%); %pi %pi %pi %pi (%o3) [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(%i4) trigeval(%); sqrt(sqrt(2) + 2) %i sqrt(2 - sqrt(2)) (%o4) [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
Categories:
Trigonometric functions
·
Package trigtools
·
99.2.7 Contract atan Functions ¶
- Function: atan_contract (r) ¶
The function atan_contract(r) contracts atan functions. We assume: \(|r| < {\pi\over 2}.\)
load("trigtools")loads this function.Examples:
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(%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 -
(%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
(%i1) load("trigtools")$(%i2) 4*atan(1/5)-atan(1/239)=atan_contract(4*atan(1/5)-atan(1/239)); 1 1 %pi (%o2) 4 atan(-) - atan(---) = --- 5 239 4 - see https://en.wikipedia.org/wiki/Machin-like_formula
(%i1) load("trigtools")$ (%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
Categories: Trigonometric functions · Package trigtools ·-