Elementary Functions - 4.42 Solution of Triangles

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.42.E1 sin ⁑ A = a c 𝐴 π‘Ž 𝑐 {\displaystyle{\displaystyle\sin A=\frac{a}{c}}}
\sin@@{A} = \frac{a}{c}		 

sin(A) = (a)/(c)

Sin[A] == Divide[a,c]
Failure Failure
Failed [300 / 300]
Result: -.1410196655+.3375964631*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -1.5}


Result: 1.858980334+.3375964631*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = 1.5}


Result: -2.141019666+.3375964631*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -.5}


Result: 3.858980334+.3375964631*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.14101966569986213, 0.33759646322287]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]}


Result: Complex[1.8589803343001379, 0.33759646322287]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]}

... skip entries to safe data
4.42.E1 a c = 1 csc ⁑ A π‘Ž 𝑐 1 𝐴 {\displaystyle{\displaystyle\frac{a}{c}=\frac{1}{\csc A}}}
\frac{a}{c} = \frac{1}{\csc@@{A}}		 

(a)/(c) = (1)/(csc(A))

Divide[a,c] == Divide[1,Csc[A]]
Failure Failure
Failed [300 / 300]
Result: .1410196654-.3375964632*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -1.5}


Result: -1.858980335-.3375964632*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = 1.5}


Result: 2.141019665-.3375964632*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -.5}


Result: -3.858980335-.3375964632*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.14101966569986213, -0.33759646322287]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]}


Result: Complex[-1.8589803343001379, -0.33759646322287]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]}

... skip entries to safe data
4.42.E2 cos ⁑ A = b c 𝐴 𝑏 𝑐 {\displaystyle{\displaystyle\cos A=\frac{b}{c}}}
\cos@@{A} = \frac{b}{c}		 

cos(A) = (b)/(c)

Cos[A] == Divide[b,c]
Failure Failure
Failed [300 / 300]
Result: -.2694569811-.3969495503*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -1.5}


Result: 1.730543019-.3969495503*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5}


Result: -2.269456981-.3969495503*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5}


Result: 3.730543019-.3969495503*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.2694569809427748, -0.3969495502290325]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]}


Result: Complex[1.730543019057225, -0.3969495502290325]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]}

... skip entries to safe data
4.42.E2 b c = 1 sec ⁑ A 𝑏 𝑐 1 𝐴 {\displaystyle{\displaystyle\frac{b}{c}=\frac{1}{\sec A}}}
\frac{b}{c} = \frac{1}{\sec@@{A}}		 

(b)/(c) = (1)/(sec(A))

Divide[b,c] == Divide[1,Sec[A]]
Failure Failure
Failed [300 / 300]
Result: .2694569810+.3969495505*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -1.5}


Result: -1.730543019+.3969495505*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5}


Result: 2.269456981+.3969495505*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5}


Result: -3.730543019+.3969495505*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.2694569809427748, 0.3969495502290325]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]}


Result: Complex[-1.730543019057225, 0.3969495502290325]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]}

... skip entries to safe data
4.42.E3 tan ⁑ A = a b 𝐴 π‘Ž 𝑏 {\displaystyle{\displaystyle\tan A=\frac{a}{b}}}
\tan@@{A} = \frac{a}{b}		 

tan(A) = (a)/(b)

Tan[A] == Divide[a,b]
Failure Failure
Failed [300 / 300]
Result: -.2860691196+.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5}


Result: 1.713930880+.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5}


Result: -2.286069120+.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5}


Result: 3.713930880+.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.2860691197539781, 0.8500402971922751]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5]}


Result: Complex[1.7139308802460218, 0.8500402971922751]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, 1.5]}

... skip entries to safe data
4.42.E3 a b = 1 cot ⁑ A π‘Ž 𝑏 1 𝐴 {\displaystyle{\displaystyle\frac{a}{b}=\frac{1}{\cot A}}}
\frac{a}{b} = \frac{1}{\cot@@{A}}		 

(a)/(b) = (1)/(cot(A))

Divide[a,b] == Divide[1,Cot[A]]
Failure Failure
Failed [300 / 300]
Result: .2860691196-.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5}


Result: -1.713930880-.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5}


Result: 2.286069120-.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5}


Result: -3.713930880-.8500402975*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.2860691197539781, -0.8500402971922751]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5]}


Result: Complex[-1.7139308802460218, -0.8500402971922751]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, 1.5]}

... skip entries to safe data
4.42.E4 a sin ⁑ A = b sin ⁑ B π‘Ž 𝐴 𝑏 𝐡 {\displaystyle{\displaystyle\frac{a}{\sin A}=\frac{b}{\sin B}}}
\frac{a}{\sin@@{A}} = \frac{b}{\sin@@{B}}		 

(a)/(sin(A)) = (b)/(sin(B))

Divide[a,Sin[A]] == Divide[b,Sin[B]]
Failure Failure
Failed [288 / 300]
Result: -3.025222791+1.188973104*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5}


Result: -1.008407597+.3963243680*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5}


Result: -2.016815194+.7926487360*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5}


Result: .5042037985-.1981621840*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -2}

... skip entries to safe data
Failed [290 / 300]
Result: Complex[-2.3601096690692955, -0.4904383214455733]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.6651131226742772, 1.6794114261511237]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.42.E4 b sin ⁑ B = c sin ⁑ C 𝑏 𝐡 𝑐 𝐢 {\displaystyle{\displaystyle\frac{b}{\sin B}=\frac{c}{\sin C}}}
\frac{b}{\sin@@{B}} = \frac{c}{\sin@@{C}}		 

(b)/(sin(B)) = (c)/(sin(C))

Divide[b,Sin[B]] == Divide[c,Sin[C]]
Failure Failure
Failed [288 / 300]
Result: -3.025222791+1.188973104*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5}


Result: -1.008407597+.3963243680*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5}


Result: -2.016815194+.7926487360*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5}


Result: .5042037985-.1981621840*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -2}

... skip entries to safe data
Failed [290 / 300]
Result: Complex[-2.3601096690692955, -0.4904383214455733]
Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.6651131226742772, 1.6794114261511237]
Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.42.E5 c 2 = a 2 + b 2 - 2 ⁒ a ⁒ b ⁒ cos ⁑ C superscript 𝑐 2 superscript π‘Ž 2 superscript 𝑏 2 2 π‘Ž 𝑏 𝐢 {\displaystyle{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C}}
c^{2} = a^{2}+b^{2}-2ab\cos@@{C}		 

(c)^(2) = (a)^(2)+ (b)^(2)- 2*a*b*cos(C)

(c)^(2) == (a)^(2)+ (b)^(2)- 2*a*b*Cos[C]
Failure Failure
Failed [300 / 300]
Result: 1.037443585-1.786272976*I
Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5}


Result: 1.037443585-1.786272976*I
Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5}


Result: -.962556415-1.786272976*I
Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5}


Result: -.962556415-1.786272976*I
Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.0374435857575133, -1.7862729760306462]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.274944825888497, 2.1108391932082666]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.42.E6 a = b ⁒ cos ⁑ C + c ⁒ cos ⁑ B π‘Ž 𝑏 𝐢 𝑐 𝐡 {\displaystyle{\displaystyle a=b\cos C+c\cos B}}
a = b\cos@@{C}+c\cos@@{B}		 

a = b*cos(C)+ c*cos(B)

a == b*Cos[C]+ c*Cos[B]
Failure Failure
Failed [300 / 300]
Result: .691629057-1.190848651*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5}


Result: -1.5
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5}


Result: -.38913962e-1-.7938991006*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5}


Result: -.7694569811-.3969495503*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6916290571716757, -1.1908486506870974]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.4374628038820034, 0.10818873905920678]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.42.E7 area = 1 2 ⁒ b ⁒ c ⁒ sin ⁑ A area 1 2 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\hbox{area}=\tfrac{1}{2}bc\sin A}}
\hbox{area} = \tfrac{1}{2}bc\sin@@{A}		 

a*r*exp(1)*a = (1)/(2)*b*c*sin(A)

a*r*E*a == Divide[1,2]*b*c*Sin[A]
Failure Failure
Failed [300 / 300]
Result: -10.14055405-.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = -1.5}


Result: 8.207848294-.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = 1.5}


Result: -4.024419932-.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = -.5}


Result: 2.091714180-.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-10.140554047136932, -0.37979602112572874]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[r, -1.5]}


Result: Complex[8.207848294961623, -0.37979602112572874]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[r, 1.5]}

... skip entries to safe data
4.42.E7 1 2 ⁒ b ⁒ c ⁒ sin ⁑ A = ( s ⁒ ( s - a ) ⁒ ( s - b ) ⁒ ( s - c ) ) 1 / 2 1 2 𝑏 𝑐 𝐴 superscript 𝑠 𝑠 π‘Ž 𝑠 𝑏 𝑠 𝑐 1 2 {\displaystyle{\displaystyle\tfrac{1}{2}bc\sin A=\left(s(s-a)(s-b)(s-c)\right)% ^{1/2}}}
\tfrac{1}{2}bc\sin@@{A} = \left(s(s-a)(s-b)(s-c)\right)^{1/2}		 

(1)/(2)*b*c*sin(A) = (s*(s - a)*(s - b)*(s - c))^(1/2)
 
Divide[1,2]*b*c*Sin[A] == (s*(s - a)*(s - b)*(s - c))^(1/2)
 Failure Failure
Failed [300 / 300]
Result: .9663528763+.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = -1.5}

Result: -5.397608155+.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = 1.5}

Result: .9663528763-.3273107602*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = -.5}

Result: -1.033647124+.3797960210*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.9663528760876551, 0.37979602112572874]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[s, -1.5]}

Result: Complex[-5.397608154591272, 0.37979602112572874]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[s, 1.5]}

... skip entries to safe data
4.42.E8 cos ⁑ a = cos ⁑ b ⁒ cos ⁑ c + sin ⁑ b ⁒ sin ⁑ c ⁒ cos ⁑ A π‘Ž 𝑏 𝑐 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\cos a=\cos b\cos c+\sin b\sin c\cos A}}
\cos@@{a} = \cos@@{b}\cos@@{c}+\sin@@{b}\sin@@{c}\cos@@{A}		 

cos(a) = cos(b)*cos(c)+ sin(b)*sin(c)*cos(A)
 
Cos[a] == Cos[b]*Cos[c]+ Sin[b]*Sin[c]*Cos[A]
 Failure Failure
Failed [300 / 300]
Result: -.6611541130+.3949633133*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5}

Result: .7926210130-.3949633133*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5}

Result: -.3407041550+.1898310285*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5}

Result: .3580230890-.1898310285*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.6611541132159315, 0.3949633132423481]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]}

Result: Complex[0.7926210131517828, -0.3949633132423481]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]}

... skip entries to safe data
4.42.E9 sin ⁑ A sin ⁑ a = sin ⁑ B sin ⁑ b 𝐴 π‘Ž 𝐡 𝑏 {\displaystyle{\displaystyle\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}}}
\frac{\sin@@{A}}{\sin@@{a}} = \frac{\sin@@{B}}{\sin@@{b}}		 

(sin(A))/(sin(a)) = (sin(B))/(sin(b))
 
Divide[Sin[A],Sin[a]] == Divide[Sin[B],Sin[b]]
 Failure Failure
Failed [288 / 300]
Result: -1.722274990-.6768885409*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5}

Result: .9305491492+.3657244397*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5}

Result: -2.652824140-1.042612981*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5}

Result: .835262737e-1+.328274973e-1*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -2}

... skip entries to safe data
Failed [286 / 300]
Result: Complex[-1.5335532645785146, 0.5223487441958409]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-0.18872172594452297, -1.1992372855051223]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.42.E9 sin ⁑ B sin ⁑ b = sin ⁑ C sin ⁑ c 𝐡 𝑏 𝐢 𝑐 {\displaystyle{\displaystyle\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}}}
\frac{\sin@@{B}}{\sin@@{b}} = \frac{\sin@@{C}}{\sin@@{c}}		 

(sin(B))/(sin(b)) = (sin(C))/(sin(c))
 
Divide[Sin[B],Sin[b]] == Divide[Sin[C],Sin[c]]
 Failure Failure
Failed [288 / 300]
Result: -1.722274990-.6768885409*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5}

Result: .9305491492+.3657244397*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5}

Result: -2.652824140-1.042612981*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5}

Result: .835262737e-1+.328274973e-1*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -2}

... skip entries to safe data
Failed [286 / 300]
Result: Complex[-1.5335532645785146, 0.5223487441958409]
Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-0.18872172594452297, -1.1992372855051223]
Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.42.E10 sin ⁑ a ⁒ cos ⁑ B = cos ⁑ b ⁒ sin ⁑ c - sin ⁑ b ⁒ cos ⁑ c ⁒ cos ⁑ A π‘Ž 𝐡 𝑏 𝑐 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A}}
\sin@@{a}\cos@@{B} = \cos@@{b}\sin@@{c}-\sin@@{b}\cos@@{c}\cos@@{A}		 

sin(a)*cos(B) = cos(b)*sin(c)- sin(b)*cos(c)*cos(A)
 
Sin[a]*Cos[B] == Cos[b]*Sin[c]- Sin[b]*Cos[c]*Cos[A]
 Failure Failure
Failed [300 / 300]
Result: -.7097001135+.4239639484*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5}

Result: -.8508201215+.4239639484*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5}

Result: -1.334305598+.7434385530*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5}

Result: -1.402132040+.7434385530*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.7097001133469564, 0.4239639481520351]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]}

Result: Complex[-0.8508201214068235, 0.4239639481520351]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]}

... skip entries to safe data
4.42.E11 cos ⁑ a ⁒ cos ⁑ C = sin ⁑ a ⁒ cot ⁑ b - sin ⁑ C ⁒ cot ⁑ B π‘Ž 𝐢 π‘Ž 𝑏 𝐢 𝐡 {\displaystyle{\displaystyle\cos a\cos C=\sin a\cot b-\sin C\cot B}}
\cos@@{a}\cos@@{C} = \sin@@{a}\cot@@{b}-\sin@@{C}\cot@@{B}		 

cos(a)*cos(C) = sin(a)*cot(b)- sin(C)*cot(B)
 
Cos[a]*Cos[C] == Sin[a]*Cot[b]- Sin[C]*Cot[B]
 Failure Failure
Failed [300 / 300]
Result: .7114823860-.4250286508*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5}

Result: .8529567893-.4250286508*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5}

Result: -1.043682738-.4250286508*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5}

Result: 2.608121914-.4250286508*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.7114823862555057, -0.42502865061548756]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.21981752916457492, 0.9933277802647092]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.42.E12 cos ⁑ A = - cos ⁑ B ⁒ cos ⁑ C + sin ⁑ B ⁒ sin ⁑ C ⁒ cos ⁑ a 𝐴 𝐡 𝐢 𝐡 𝐢 π‘Ž {\displaystyle{\displaystyle\cos A=-\cos B\cos C+\sin B\sin C\cos a}}
\cos@@{A} = -\cos@@{B}\cos@@{C}+\sin@@{B}\sin@@{C}\cos@@{a}		 

cos(A) = - cos(B)*cos(C)+ sin(B)*sin(C)*cos(a)
 
Cos[A] == - Cos[B]*Cos[C]+ Sin[B]*Sin[C]*Cos[a]
 Failure Failure
Failed [300 / 300]
Result: 1.062535945-1.017952978*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5}

Result: 1.062535945-1.017952978*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = 1.5}

Result: .5591646152-1.485905089*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -.5}

Result: .5591646152-1.485905089*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.0625359450203713, -1.017952977441946]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.8749374794081675, -0.5777856599721184]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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