Julian D. A. Wiseman
Contents: values of Tan[α] expressed in surds, for α=n×3° or α=n×5⅝°, n∈ℕ: Tan[0°], Tan[3°], Tan[5.625°], Tan[6°], Tan[9°], Tan[11.25°], Tan[12°], Tan[15°], Tan[16.875°], Tan[18°], Tan[21°], Tan[22.5°], Tan[24°], Tan[27°], Tan[28.125°], Tan[30°], Tan[33°], Tan[33.75°], Tan[36°], Tan[39°], Tan[39.375°], Tan[42°], Tan[45°], Tan[48°], Tan[50.625°], Tan[51°], Tan[54°], Tan[56.25°], Tan[57°], Tan[60°], Tan[61.875°], Tan[63°], Tan[66°], Tan[67.5°], Tan[69°], Tan[72°], Tan[73.125°], Tan[75°], Tan[78°], Tan[78.75°], Tan[81°], Tan[84°], Tan[84.375°], Tan[87°], Tan[90°].
Publication history: only here. Usual disclaimer and copyright terms apply. Also see the values of Sin[] and Cos[], in surds, the values of Cosecant[] = Cosec[] = Csc[] = 1/Sin[] and Secant[] = Sec[] = 1/Cos[], in surds, and the inner radius of n/m stars, in surds.
The table shows Tan[] in surds, for angles that are integer multiples of 3° or of 5⅝° = 90°/16. The surds are shown in several formats.
Graphical formula: a .png, derived from…
LaTex: a LaTeX expression.
Excel: copy-pasteable into Excel, which will automatically convert the “Sqrt” into an upper-case “SQRT”.
CalcCenter: if one enters Sqrt[5-2 Sqrt[5]] directly into Mathematica CalcCenter (the budget version of Mathematica), it automatically evaluates the expression numerically, frustrating an attempt to work with surds. To prevent this integers inside the inner-most Sqrts have been replaced with the likes of Int5, which CalcCenter treats as a variable. If using the full-expense Mathematica instances of “Int” may be removed by a preprocessor or by setting Int2=2; Int3=3; Int5=5;.
Postscript: using 5 5 sqrt 2 mul sub sqrt rather than 36 dup sin exch cos div is slightly more efficient computationally, but is also—more importantly—far more elegant.
Help! Further simplifications are wanted, and credit will be given to the sender.
Errors: whilst the outputs have been tested, it is possible that errors remain. Please do test things before embedding them somewhere important—and if errors or possible improvements are found, tell the author.
| Tan[α] = 1/Tan[90–α] | Graphical formula | LaTeX | Excel | CalcCenter | PostScript |
|---|---|---|---|---|---|
| Tan[0°] = 1/Tan[90°] = 0 | Tan(0°) | 0 | 0 | 0 | 0 |
| Tan[3°] = 1/Tan[87°] ≈ 0.052407779283 | [画像:Tan(3°)] | \frac{1}{4} \left(1 + 2 \sqrt{5 - 2 \sqrt{5}} - \sqrt{5}\right) \left(\left(\sqrt{5} + 2\right) \left(2 \sqrt{3} - 3\right) - 1\right) | =( 1 + 2*Sqrt(5-2*Sqrt(5)) - Sqrt(5) ) * ((Sqrt(5)+2)*(2*Sqrt(3)-3)-1) / 4 | ( 1 + 2 Sqrt[5-2 Sqrt[Int5]] - Sqrt[Int5] ) ((Sqrt[Int5]+2) (2 Sqrt[Int3]-3)-1) / 4 | 5 sqrt dup dup -2 mul 5 add sqrt 2 mul 1 add exch sub exch 2 add 3 sqrt 2 mul 3 sub mul 1 sub mul 4 div |
| Tan[5⅝°] = 1/Tan[84⅜°] ≈ 0.098491403357 | [画像:Tan(5.625°)] | \frac{1}{2} \left(2 - \sqrt{2 + \sqrt{2 + \sqrt{2}}}\right) \sqrt{2 \left(2 + \sqrt{2}\right) \left(2 + \sqrt{2 + \sqrt{2}}\right)} | =( 2 - Sqrt(2+Sqrt(2+Sqrt(2))) ) * Sqrt( 2 * (2+Sqrt(2)) * (2+Sqrt(2+Sqrt(2))) ) / 2 | ( 2 - Sqrt[2+Sqrt[2+Sqrt[Int2]]] ) Sqrt[ 2 (2+Sqrt[Int2]) (2+Sqrt[2+Sqrt[Int2]]) ] / 2 | 2 sqrt 2 add dup sqrt 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 sub neg mul 2 div |
| Tan[6°] = 1/Tan[84°] ≈ 0.105104235266 | [画像:Tan(6°)] | \frac{1}{2} \left(\sqrt{10 - 2 \sqrt{5}} - \sqrt{3} \left(\sqrt{5} - 1\right)\right) | =( Sqrt(10-2*Sqrt(5)) - Sqrt(3)*(Sqrt(5)-1) ) / 2 | ( Sqrt[10-2 Sqrt[Int5]] - Sqrt[Int3] (Sqrt[Int5]-1) ) / 2 | 5 sqrt dup -2 mul 10 add sqrt exch 1 sub 3 sqrt mul sub 2 div |
| Tan[9°] = 1/Tan[81°] ≈ 0.158384440325 | Tan(9°) | 1 + \sqrt{5} - \sqrt{2 \sqrt{5} + 5} | =1 + Sqrt(5) - Sqrt(2*Sqrt(5)+5) | 1 + Sqrt[Int5] - Sqrt[2 Sqrt[Int5]+5] | 5 sqrt dup 1 add exch 2 mul 5 add sqrt sub |
| Tan[11¼°] = 1/Tan[78¾°] ≈ 0.19891236738 | [画像:Tan(11.25°)] | \left(\sqrt{\sqrt{2} + 2} - 1\right) \sqrt{2} - 1 | =(Sqrt(Sqrt(2)+2)-1)*Sqrt(2) - 1 | (Sqrt[Sqrt[Int2]+2]-1) Sqrt[Int2] - 1 | 2 sqrt dup 2 add sqrt 1 sub mul 1 sub |
| Tan[12°] = 1/Tan[78°] ≈ 0.21255656167 | [画像:Tan(12°)] | \frac{1}{4} \left(2 \sqrt{3} - \sqrt{10 - 2 \sqrt{5}}\right) \left(3 - \sqrt{5}\right) | =(2*Sqrt(3) - Sqrt(10-2*Sqrt(5))) * (3-Sqrt(5)) / 4 | (2 Sqrt[Int3] - Sqrt[10-2 Sqrt[Int5]]) (3-Sqrt[Int5]) / 4 | 5 sqrt neg dup 2 mul 10 add sqrt neg 3 sqrt 2 mul add exch 3 add mul 4 div |
| Tan[15°] = 1/Tan[75°] ≈ 0.267949192431 | Tan(15°) | 2 - \sqrt{3} | =2-Sqrt(3) | 2-Sqrt[Int3] | 2 3 sqrt sub |
| Tan[16⅞°] = 1/Tan[73⅛°] ≈ 0.303346683607 | [画像:Tan(16.875°)] | \frac{1}{2} \left(2 - \sqrt{2 + \sqrt{2 - \sqrt{2}}}\right) \sqrt{2 \left(2 - \sqrt{2}\right) \left(2 + \sqrt{2 - \sqrt{2}}\right)} | =( 2 - Sqrt(2+Sqrt(2-Sqrt(2))) ) * Sqrt( 2 * (2-Sqrt(2)) * (2+Sqrt(2-Sqrt(2))) ) / 2 | ( 2 - Sqrt[2+Sqrt[2-Sqrt[Int2]]] ) Sqrt[ 2 (2-Sqrt[Int2]) (2+Sqrt[2-Sqrt[Int2]]) ] / 2 | 2 2 sqrt sub dup sqrt 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 sub neg mul 2 div |
| Tan[18°] = 1/Tan[72°] ≈ 0.324919696233 | [画像:Tan(18°)] | \sqrt{1 - \frac{2}{5}\sqrt{5}} | =Sqrt(1-Sqrt(5)*2/5) | Sqrt[1-Sqrt[Int5] 2/5] | 1 5 sqrt 2 mul 5 div sub sqrt |
| Tan[21°] = 1/Tan[69°] ≈ 0.383864035035 | [画像:Tan(21°)] | \frac{1}{4} \left(2 \sqrt{5 - 2 \sqrt{5}} + \sqrt{5} - 3\right) \left(1 + 2 \sqrt{3} - \sqrt{5}\right) | =(2*Sqrt(5-2*Sqrt(5)) + Sqrt(5) - 3) * (1 + 2*Sqrt(3) - Sqrt(5)) / 4 | (2 Sqrt[5-2 Sqrt[Int5]] + Sqrt[Int5] - 3) (1 + 2 Sqrt[Int3] - Sqrt[Int5]) / 4 | 5 sqrt dup dup -2 mul 5 add sqrt 2 mul add 3 sub exch neg 3 sqrt 2 mul add 1 add mul 4 div |
| Tan[22½°] = 1/Tan[67½°] ≈ 0.414213562373 | Tan(22.5°) | \sqrt{2} - 1 | =Sqrt(2)-1 | Sqrt[Int2]-1 | 2 sqrt 1 sub |
| Tan[24°] = 1/Tan[66°] ≈ 0.445228685309 | [画像:Tan(24°)] | \frac{1}{2} \left(\sqrt{22 \sqrt{5} + 50} - \sqrt{3} \left(\sqrt{5} + 3\right)\right) | =( Sqrt(22*Sqrt(5)+50) - Sqrt(3)*(Sqrt(5)+3) ) / 2 | ( Sqrt[22 Sqrt[Int5]+50] - Sqrt[Int3] (Sqrt[Int5]+3) ) / 2 | 5 sqrt dup 22 mul 50 add sqrt exch 3 add 3 sqrt mul sub 2 div |
| Tan[27°] = 1/Tan[63°] ≈ 0.509525449494 | Tan(27°) | \sqrt{5} - 1 - \sqrt{5 - 2 \sqrt{5}} | =Sqrt(5) - 1 - Sqrt(5-2*Sqrt(5)) | Sqrt[Int5] - 1 - Sqrt[5-2 Sqrt[Int5]] | 5 sqrt dup -2 mul 5 add sqrt sub 1 sub |
| Tan[28⅛°] = 1/Tan[61⅞°] ≈ 0.534511135951 | [画像:Tan(28.125°)] | \frac{1}{2} \left(2 - \sqrt{2 - \sqrt{2 - \sqrt{2}}}\right) \sqrt{2 \left(2 - \sqrt{2}\right) \left(2 - \sqrt{2 - \sqrt{2}}\right)} | =( 2 - Sqrt(2-Sqrt(2-Sqrt(2))) ) * Sqrt( 2 * (2-Sqrt(2)) * (2-Sqrt(2-Sqrt(2))) ) / 2 | ( 2 - Sqrt[2-Sqrt[2-Sqrt[Int2]]] ) Sqrt[ 2 (2-Sqrt[Int2]) (2-Sqrt[2-Sqrt[Int2]]) ] / 2 | 2 2 sqrt sub dup sqrt neg 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt neg 2 add mul 2 div |
| Tan[30°] = 1/Tan[60°] ≈ 0.57735026919 | Tan(30°) | \frac{\sqrt{3}}{3} | =Sqrt(3)/3 | Sqrt[Int3]/3 | 3 sqrt 3 div |
| Tan[33°] = 1/Tan[57°] ≈ 0.649407593198 | [画像:Tan(33°)] | \frac{1}{2} \left(\sqrt{5 - \sqrt{5}} + \sqrt{2}\right) \sqrt{\frac{1}{5} \left(8 - \left(5 \sqrt{5} + 11\right) \left(10 \sqrt{3} - 17\right)\right)} | =(Sqrt(5-Sqrt(5))+Sqrt(2)) * Sqrt( ( 8-(5*Sqrt(5)+11)*(10*Sqrt(3)-17) ) / 5 ) / 2 | (Sqrt[5-Sqrt[Int5]]+Sqrt[Int2]) Sqrt[ ( 8-(5 Sqrt[Int5]+11) (10 Sqrt[Int3]-17) ) / Int5 ] / 2 | 5 sqrt dup neg 5 add sqrt 2 sqrt add exch 5 mul 11 add 17 3 sqrt 10 mul sub mul 8 add 5 div sqrt mul 2 div |
| Tan[33¾°] = 1/Tan[56¼°] ≈ 0.668178637919 | [画像:Tan(33.75°)] | \left(\sqrt{2 - \sqrt{2}} - 1\right) \sqrt{2} + 1 | =(Sqrt(2-Sqrt(2))-1)*Sqrt(2) + 1 | (Sqrt[2-Sqrt[Int2]]-1) Sqrt[Int2] + 1 | 2 sqrt dup 2 sub neg sqrt 1 sub mul 1 add |
| Tan[36°] = 1/Tan[54°] ≈ 0.726542528005 | Tan(36°) | \sqrt{5 - 2 \sqrt{5}} | =Sqrt(5-2*Sqrt(5)) | Sqrt[5-2 Sqrt[Int5]] | 5 5 sqrt 2 mul sub sqrt |
| Tan[39°] = 1/Tan[51°] ≈ 0.809784033195 | [画像:Tan(39°)] | \frac{1}{2} \left(\sqrt{5 + \sqrt{5}} - \sqrt{2}\right) \sqrt{\frac{1}{5} \left(8 + \left(5 \sqrt{5} - 11\right) \left(10 \sqrt{3} - 17\right)\right)} | =(Sqrt(5+Sqrt(5))-Sqrt(2)) * Sqrt( ( 8+(5*Sqrt(5)-11)*(10*Sqrt(3)-17) ) / 5 ) / 2 | (Sqrt[5+Sqrt[Int5]]-Sqrt[Int2]) Sqrt[ ( 8+(5 Sqrt[Int5]-11) (10 Sqrt[Int3]-17) ) / Int5 ] / 2 | 5 sqrt dup 5 add sqrt 2 sqrt sub exch 5 mul 11 sub 3 sqrt 10 mul 17 sub mul 8 add 5 div sqrt mul 2 div |
| Tan[39⅜°] = 1/Tan[50⅝°] ≈ 0.820678790829 | [画像:Tan(39.375°)] | \frac{1}{2} \left(2 - \sqrt{2 - \sqrt{2 + \sqrt{2}}}\right) \sqrt{2 \left(2 + \sqrt{2}\right) \left(2 - \sqrt{2 + \sqrt{2}}\right)} | =( 2 - Sqrt(2-Sqrt(2+Sqrt(2))) ) * Sqrt( 2 * (2+Sqrt(2)) * (2-Sqrt(2+Sqrt(2))) ) / 2 | ( 2 - Sqrt[2-Sqrt[2+Sqrt[Int2]]] ) Sqrt[ 2 (2+Sqrt[Int2]) (2-Sqrt[2+Sqrt[Int2]]) ] / 2 | 2 2 sqrt add dup sqrt neg 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt neg 2 add mul 2 div |
| Tan[42°] = 1/Tan[48°] ≈ 0.900404044298 | [画像:Tan(42°)] | \frac{1}{4} \left(2 \sqrt{3} - \sqrt{10 - 2 \sqrt{5}}\right) \left(1 + \sqrt{5}\right) | =(2*Sqrt(3)-Sqrt(10-2*Sqrt(5))) * (1+Sqrt(5)) / 4 | (2 Sqrt[Int3]-Sqrt[10-2 Sqrt[Int5]]) (1+Sqrt[Int5]) / 4 | 5 sqrt dup -2 mul 10 add sqrt neg 3 sqrt 2 mul add exch 1 add mul 4 div |
| Tan[45°] = 1 | Tan(45°) | 1 | 1 | 1 | 1 |
| Tan[48°] = 1/Tan[42°] ≈ 1.110612514829 | [画像:Tan(48°)] | \frac{1}{4} \left(2 \sqrt{3} + \sqrt{10 - 2 \sqrt{5}}\right) \left(3 - \sqrt{5}\right) | =(2*Sqrt(3)+Sqrt(10-2*Sqrt(5))) * (3-Sqrt(5)) / 4 | (2 Sqrt[Int3]+Sqrt[10-2 Sqrt[Int5]]) (3-Sqrt[Int5]) / 4 | 5 sqrt neg dup 2 mul 10 add sqrt 3 sqrt 2 mul add exch 3 add mul 4 div |
| Tan[50⅝°] = 1/Tan[39⅜°] ≈ 1.218503525588 | [画像:Tan(50.625°)] | \frac{1}{2} \left(2 + \sqrt{2 - \sqrt{2 + \sqrt{2}}}\right) \sqrt{2 \left(2 + \sqrt{2}\right) \left(2 - \sqrt{2 + \sqrt{2}}\right)} | =( 2 + Sqrt(2-Sqrt(2+Sqrt(2))) ) * Sqrt( 2 * (2+Sqrt(2)) * (2-Sqrt(2+Sqrt(2))) ) / 2 | ( 2 + Sqrt[2-Sqrt[2+Sqrt[Int2]]] ) Sqrt[ 2 (2+Sqrt[Int2]) (2-Sqrt[2+Sqrt[Int2]]) ] / 2 | 2 sqrt 2 add dup sqrt neg 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 add mul 2 div |
| Tan[51°] = 1/Tan[39°] ≈ 1.234897156535 | [画像:Tan(51°)] | \frac{1}{2} \left(\sqrt{5 + \sqrt{5}} + \sqrt{2}\right) \sqrt{\frac{1}{5} \left(8 - \left(5 \sqrt{5} - 11\right) \left(10 \sqrt{3} + 17\right)\right)} | =(Sqrt(5+Sqrt(5))+Sqrt(2)) * Sqrt( ( 8-(5*Sqrt(5)-11)*(10*Sqrt(3)+17) ) / 5 ) / 2 | (Sqrt[5+Sqrt[Int5]]+Sqrt[Int2]) Sqrt[ ( 8-(5 Sqrt[Int5]-11) (10 Sqrt[Int3]+17) ) / Int5 ] / 2 | 5 sqrt dup 5 add sqrt 2 sqrt add exch 5 mul 11 sub 3 sqrt 10 mul 17 add mul neg 8 add 5 div sqrt mul 2 div |
| Tan[54°] = 1/Tan[36°] ≈ 1.376381920471 | [画像:Tan(54°)] | \sqrt{1 + \frac{2}{5}\sqrt{5}} | =Sqrt(1+Sqrt(5)*2/5) | Sqrt[1+Sqrt[Int5] 2/5] | 5 sqrt 2 mul 5 div 1 add sqrt |
| Tan[56¼°] = 1/Tan[33¾°] ≈ 1.496605762665 | [画像:Tan(56.25°)] | \left(\sqrt{2 - \sqrt{2}} + 1\right) \sqrt{2} - 1 | =(Sqrt(2-Sqrt(2))+1)*Sqrt(2) - 1 | (Sqrt[2-Sqrt[Int2]]+1) Sqrt[Int2] - 1 | 2 sqrt dup 2 sub neg sqrt 1 add mul 1 sub |
| Tan[57°] = 1/Tan[33°] ≈ 1.539864963815 | [画像:Tan(57°)] | \frac{1}{2} \left(\sqrt{5 - \sqrt{5}} - \sqrt{2}\right) \sqrt{\frac{1}{5} \left(8 + \left(5 \sqrt{5} + 11\right) \left(10 \sqrt{3} + 17\right)\right)} | =(Sqrt(5-Sqrt(5))-Sqrt(2)) * Sqrt( ( 8+(5*Sqrt(5)+11)*(10*Sqrt(3)+17) ) / 5 ) / 2 | (Sqrt[5-Sqrt[Int5]]-Sqrt[Int2]) Sqrt[ ( 8+(5 Sqrt[Int5]+11) (10 Sqrt[Int3]+17) ) / Int5 ] / 2 | 5 sqrt dup neg 5 add sqrt 2 sqrt sub exch 5 mul 11 add 17 3 sqrt 10 mul add mul 8 add 5 div sqrt mul 2 div |
| Tan[60°] = 1/Tan[30°] ≈ 1.732050807569 | Tan(60°) | \sqrt{3} | =Sqrt(3) | Sqrt[Int3] | 3 sqrt |
| Tan[61⅞°] = 1/Tan[28⅛°] ≈ 1.870868411789 | [画像:Tan(61.875°)] | \frac{1}{2} \left(2 + \sqrt{2 - \sqrt{2 - \sqrt{2}}}\right) \sqrt{2 \left(2 - \sqrt{2}\right) \left(2 - \sqrt{2 - \sqrt{2}}\right)} | =( 2 + Sqrt(2-Sqrt(2-Sqrt(2))) ) * Sqrt( 2 * (2-Sqrt(2)) * (2-Sqrt(2-Sqrt(2))) ) / 2 | ( 2 + Sqrt[2-Sqrt[2-Sqrt[Int2]]] ) Sqrt[ 2 (2-Sqrt[Int2]) (2-Sqrt[2-Sqrt[Int2]]) ] / 2 | 2 2 sqrt sub dup sqrt neg 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 add mul 2 div |
| Tan[63°] = 1/Tan[27°] ≈ 1.962610505505 | Tan(63°) | \sqrt{5 - 2 \sqrt{5}} + \sqrt{5} - 1 | =Sqrt(5-2*Sqrt(5)) + Sqrt(5) - 1 | Sqrt[5-2 Sqrt[Int5]] + Sqrt[Int5] - 1 | 5 sqrt dup -2 mul 5 add sqrt add 1 sub |
| Tan[66°] = 1/Tan[24°] ≈ 2.246036773904 | [画像:Tan(66°)] | \frac{1}{2} \left(\sqrt{3} \left(\sqrt{5} - 1\right) + \sqrt{10 - 2 \sqrt{5}}\right) | =( Sqrt(3)*(Sqrt(5)-1) + Sqrt(10-2*Sqrt(5)) ) / 2 | ( Sqrt[Int3] (Sqrt[Int5]-1) + Sqrt[10-2 Sqrt[Int5]] ) / 2 | 5 sqrt dup -2 mul 10 add sqrt exch 1 sub 3 sqrt mul add 2 div |
| Tan[67½°] = 1/Tan[22½°] ≈ 2.414213562373 | Tan(67.5°) | \sqrt{2} + 1 | =Sqrt(2)+1 | Sqrt[Int2]+1 | 2 sqrt 1 add |
| Tan[69°] = 1/Tan[21°] ≈ 2.605089064694 | [画像:Tan(69°)] | \frac{1}{2} \left(\sqrt{5 + \sqrt{5}} + \sqrt{2}\right) \sqrt{\frac{1}{5} \left(\left(5 \sqrt{5} - 11\right) \left(10 \sqrt{3} - 17\right) + 8\right)} | =(Sqrt(5+Sqrt(5))+Sqrt(2)) * Sqrt( ( (5*Sqrt(5)-11)*(10*Sqrt(3)-17)+8 ) / 5 ) / 2 | (Sqrt[5+Sqrt[Int5]]+Sqrt[Int2]) Sqrt[ ( (5 Sqrt[Int5]-11) (10 Sqrt[Int3]-17)+8 ) / Int5 ] / 2 | 5 sqrt dup 5 add sqrt 2 sqrt add exch 5 mul 11 sub 3 sqrt 10 mul 17 sub mul 8 add 5 div sqrt mul 2 div |
| Tan[72°] = 1/Tan[18°] ≈ 3.077683537175 | Tan(72°) | \sqrt{2 \sqrt{5} + 5} | =Sqrt(2*Sqrt(5)+5) | Sqrt[2 Sqrt[Int5]+5] | 5 sqrt 2 mul 5 add sqrt |
| Tan[73⅛°] = 1/Tan[16⅞°] ≈ 3.296558208938 | [画像:Tan(73.125°)] | \frac{1}{2} \left(2 + \sqrt{2 + \sqrt{2 - \sqrt{2}}}\right) \sqrt{2 \left(2 - \sqrt{2}\right) \left(2 + \sqrt{2 - \sqrt{2}}\right)} | =( 2 + Sqrt(2+Sqrt(2-Sqrt(2))) ) * Sqrt( 2 * (2-Sqrt(2)) * (2+Sqrt(2-Sqrt(2))) ) / 2 | ( 2 + Sqrt[2+Sqrt[2-Sqrt[Int2]]] ) Sqrt[ 2 (2-Sqrt[Int2]) (2+Sqrt[2-Sqrt[Int2]]) ] / 2 | 2 2 sqrt sub dup sqrt 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 add mul 2 div |
| Tan[75°] = 1/Tan[15°] ≈ 3.732050807569 | Tan(75°) | 2 + \sqrt{3} | =2+Sqrt(3) | 2+Sqrt[Int3] | 3 sqrt 2 add |
| Tan[78°] = 1/Tan[12°] ≈ 4.704630109478 | [画像:Tan(78°)] | \frac{1}{2} \left(\sqrt{3} \left(\sqrt{5} + 1\right) + \sqrt{2 \sqrt{5} + 10}\right) | =( Sqrt(3)*(Sqrt(5)+1) + Sqrt(2*Sqrt(5)+10) ) / 2 | ( Sqrt[Int3] (Sqrt[Int5]+1) + Sqrt[2 Sqrt[Int5]+10] ) / 2 | 5 sqrt dup 2 mul 10 add sqrt exch 1 add 3 sqrt mul add 2 div |
| Tan[78¾°] = 1/Tan[11¼°] ≈ 5.027339492126 | [画像:Tan(78.75°)] | \left(\sqrt{2 + \sqrt{2}} + 1\right) \sqrt{2} + 1 | =(Sqrt(2+Sqrt(2))+1)*Sqrt(2) + 1 | (Sqrt[2+Sqrt[Int2]]+1) Sqrt[Int2] + 1 | 2 sqrt dup 2 add sqrt 1 add mul 1 add |
| Tan[81°] = 1/Tan[9°] ≈ 6.313751514675 | Tan(81°) | 1 + \sqrt{5} + \sqrt{2 \sqrt{5} + 5} | =1 + Sqrt(5) + Sqrt(2*Sqrt(5)+5) | 1 + Sqrt[Int5] + Sqrt[2 Sqrt[Int5]+5] | 5 sqrt dup 1 add exch 2 mul 5 add sqrt add |
| Tan[84°] = 1/Tan[6°] ≈ 9.514364454223 | [画像:Tan(84°)] | \frac{1}{2} \left(\sqrt{3} \left(\sqrt{5} + 3\right) + \sqrt{22 \sqrt{5} + 50}\right) | =( Sqrt(3)*(Sqrt(5)+3) + Sqrt(22*Sqrt(5)+50) ) / 2 | ( Sqrt[Int3] (Sqrt[Int5]+3) + Sqrt[22 Sqrt[Int5]+50] ) / 2 | 5 sqrt dup 22 mul 50 add sqrt exch 3 add 3 sqrt mul add 2 div |
| Tan[84⅜°] = 1/Tan[5⅝°] ≈ 10.153170387609 | [画像:Tan(84.375°)] | \frac{1}{2} \left(2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}\right) \sqrt{2 \left(2 + \sqrt{2}\right) \left(2 + \sqrt{2 + \sqrt{2}}\right)} | =( 2 + Sqrt(2+Sqrt(2+Sqrt(2))) ) * Sqrt( 2 * (2+Sqrt(2)) * (2+Sqrt(2+Sqrt(2))) ) / 2 | ( 2 + Sqrt[2+Sqrt[2+Sqrt[Int2]]] ) Sqrt[ 2 (2+Sqrt[Int2]) (2+Sqrt[2+Sqrt[Int2]]) ] / 2 | 2 sqrt 2 add dup sqrt 2 add dup 3 1 roll mul 2 mul sqrt exch sqrt 2 add mul 2 div |
| Tan[87°] = 1/Tan[3°] ≈ 19.081136687728 | [画像:Tan(87°)] | \frac{1}{4} \left(2 \sqrt{5 - 2 \sqrt{5}} + \sqrt{5} - 1\right) \left(\left(2 \sqrt{3} + 3\right) \left(\sqrt{5} + 2\right) + 1\right) | =(2*Sqrt(5-2*Sqrt(5)) + Sqrt(5) - 1) * ((2*Sqrt(3)+3)*(Sqrt(5)+2)+1) / 4 | (2 Sqrt[5-2 Sqrt[Int5]] + Sqrt[Int5] - 1) ((2 Sqrt[Int3]+3) (Sqrt[Int5]+2)+1) / 4 | 5 sqrt dup dup -2 mul 5 add sqrt 2 mul add 1 sub exch 2 add 3 sqrt 2 mul 3 add mul 1 add mul 4 div |
| Tan[90°] = 1/Tan[0°] = ±∞ | Tan(90°) | \pm\infty | =(2^53 - 1) * (2^971) | ±Infinity | 2 23 exp 1 sub 2 104 exp mul |
Julian D. A. Wiseman, December 2008
