# Sin and Cos in Surds

Julian D. A. Wiseman

Contents: values of Sin[α] and Cos[α], expressed in surds, for α=n×3° or α=n×5⅝°, n∈ℕ: Sin[0°], Sin[3°], Sin[5.625°], Sin[6°], Sin[9°], Sin[11.25°], Sin[12°], Sin[15°], Sin[16.875°], Sin[18°], Sin[21°], Sin[22.5°], Sin[24°], Sin[27°], Sin[28.125°], Sin[30°], Sin[33°], Sin[33.75°], Sin[36°], Sin[39°], Sin[39.375°], Sin[42°], Sin[45°], Sin[48°], Sin[50.625°], Sin[51°], Sin[54°], Sin[56.25°], Sin[57°], Sin[60°], Sin[61.875°], Sin[63°], Sin[66°], Sin[67.5°], Sin[69°], Sin[72°], Sin[73.125°], Sin[75°], Sin[78°], Sin[78.75°], Sin[81°], Sin[84°], Sin[84.375°], Sin[87°], and Sin[90°].

Publication history: only here. Usual disclaimer and copyright terms apply. Also see the values of Cosecant[] = Cosec[] = Csc[] = 1/Sin[] and Secant[] = Sec[] = 1/Cos[], in surds, the values of Tan[] in surds, and the inner radius of n/m stars, in surds.

The table shows Sin[] and Cos[] 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[2]/2 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 sub 8 div sqrt rather than 36 sin is about as efficient computationally (well, it takes about twice the time, but who’s counting?), but is, in some sense, far more elegant.

Help! Sin and Cos of 3°, 21°, 33°, 39°, 51°, 57°, 69°, and 87° each have two answers. One is longer, but the Sqrt’s are nested only two deep; one is more concise but the Sqrt’s are three deep. A concise two-deep answer would be preferred, and other simplifications are also welcomed. Credit will be given.

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.

Credit: in this post (on a password-protected bulletin board) Arthur L. Rubin suggested a method of simplifying Sin[6°]. The method was general and quite obvious, but not quite so obvious that the author thought of it first.

Sin[α] = Cos[90–α]Graphical formulaLaTeXExcelCalcCenterPostScript
Sin[0°] = Cos[90°] = 00000
Sin[3°] = Cos[87°] ≈ 0.052335956243\frac{1}{16} \left(\sqrt{2} \left(\sqrt{3} + 1\right) \left(\sqrt{5} - 1\right) - 2 \left(\sqrt{3} - 1\right) \sqrt{\sqrt{5} + 5}\right)=( Sqrt(2)*(Sqrt(3)+1)*(Sqrt(5)-1) - 2*(Sqrt(3)-1)*Sqrt(Sqrt(5)+5) ) / 16( Sqrt[Int2] (Sqrt[Int3]+1) (Sqrt[Int5]-1) - 2 (Sqrt[Int3]-1) Sqrt[Sqrt[Int5]+5] ) / 163 sqrt 5 sqrt 2 copy 1 sub exch 1 add mul 2 sqrt mul exch 5 add sqrt 3 -1 roll 1 sub mul 2 mul sub 16 div
\frac{1}{4} \sqrt{8 - \sqrt{10 - 2 \sqrt{5}} - \sqrt{3} \left(1 + \sqrt{5}\right)}=Sqrt( 8 - Sqrt(10-2*Sqrt(5)) - Sqrt(3)*(1+Sqrt(5)) ) / 4Sqrt[ 8 - Sqrt[10-2 Sqrt[Int5]] - Sqrt[Int3] (1+Sqrt[Int5]) ] / 48 5 sqrt dup 1 add 3 sqrt mul exch -2 mul 10 add sqrt add sub sqrt 4 div
Sin[5⅝°] = Cos[84⅜°] ≈ 0.09801714033\frac{1}{2} \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2}}}}=Sqrt( 2 - Sqrt( 2 + Sqrt(2+Sqrt(2)) ) ) / 2Sqrt[ 2 - Sqrt[ 2 + Sqrt[2+Sqrt[Int2]] ] ] / 22 2 sqrt 2 add sqrt 2 add sqrt sub sqrt 2 div
Sin[6°] = Cos[84°] ≈ 0.104528463268\frac{1}{8} \left(\sqrt{30 - 6 \sqrt{5}} - 1 - \sqrt{5}\right)=( Sqrt(30-6*Sqrt(5)) - 1 - Sqrt(5) ) / 8( Sqrt[30-6 Sqrt[Int5]] - 1 - Sqrt[Int5] ) / 85 sqrt dup -6 mul 30 add sqrt exch sub 1 sub 8 div
Sin[9°] = Cos[81°] ≈ 0.15643446504\frac{1}{8} \left(\sqrt{2} \left(1 + \sqrt{5}\right) - 2 \sqrt{5 - \sqrt{5}}\right)=( Sqrt(2)*(1+Sqrt(5)) - 2*Sqrt(5-Sqrt(5)) ) / 8( Sqrt[Int2] (1+Sqrt[Int5]) - 2 Sqrt[5-Sqrt[Int5]] ) / 85 sqrt dup 1 add 2 sqrt mul exch 5 exch sub sqrt 2 mul sub 8 div
Sin[11¼°] = Cos[78¾°] ≈ 0.195090322016\frac{1}{2} \sqrt{2 - \sqrt{2 + \sqrt{2}}}=Sqrt( 2 - Sqrt(2+Sqrt(2)) ) / 2Sqrt[ 2 - Sqrt[2+Sqrt[Int2]] ] / 22 2 sqrt 2 add sqrt sub sqrt 2 div
Sin[12°] = Cos[78°] ≈ 0.207911690818\frac{1}{8} \left(\sqrt{10 + 2 \sqrt{5}} - \sqrt{3} \left(\sqrt{5} - 1\right)\right)=( Sqrt(10+2*Sqrt(5)) - Sqrt(3)*(Sqrt(5)-1) ) / 8( Sqrt[10+2 Sqrt[Int5]] - Sqrt[Int3] (Sqrt[Int5]-1) ) / 85 sqrt dup 2 mul 10 add sqrt exch 1 sub 3 sqrt mul sub 8 div
Sin[15°] = Cos[75°] ≈ 0.258819045103\frac{\sqrt{2} }{4}\left(\sqrt{3} - 1\right)=Sqrt(2) * (Sqrt(3)-1) / 4Sqrt[Int2] (Sqrt[Int3]-1) / 43 sqrt 1 sub 2 sqrt mul 4 div
Sin[16⅞°] = Cos[73⅛°] ≈ 0.290284677254\frac{1}{2} \sqrt{2 - \sqrt{2 + \sqrt{2 - \sqrt{2}}}}=Sqrt( 2 - Sqrt( 2 + Sqrt(2-Sqrt(2)) ) ) / 2Sqrt[ 2 - Sqrt[ 2 + Sqrt[2-Sqrt[Int2]] ] ] / 22 2 2 sqrt sub sqrt 2 add sqrt sub sqrt 2 div
Sin[18°] = Cos[72°] ≈ 0.309016994375\frac{1}{4} \left(\sqrt{5} - 1\right)=(Sqrt(5)-1)/4(Sqrt[Int5]-1)/45 sqrt 1 sub 4 div
Sin[21°] = Cos[69°] ≈ 0.358367949545\frac{1}{16} \left(\sqrt{3} + 1\right) \left(2 \sqrt{5 - \sqrt{5}} - \sqrt{2} \left(2 - \sqrt{3}\right) \left(1 + \sqrt{5}\right)\right)=(Sqrt(3)+1) * ( 2*Sqrt(5-Sqrt(5)) - Sqrt(2)*(2-Sqrt(3))*(1+Sqrt(5)) ) / 16(Sqrt[Int3]+1) ( 2 Sqrt[5-Sqrt[Int5]] - Sqrt[Int2] (2-Sqrt[Int3]) (1+Sqrt[Int5]) ) / 163 sqrt 5 sqrt 2 copy 1 add exch 2 sub mul 2 sqrt mul exch 5 sub neg sqrt 2 mul add exch 1 add mul 16 div
\frac{1}{4} \sqrt{8 - \sqrt{3} \left(\sqrt{5} - 1\right) - \sqrt{10 + 2 \sqrt{5}}}=Sqrt( 8 - Sqrt(3)*(Sqrt(5) - 1) - Sqrt(10 + 2*Sqrt(5)) ) / 4Sqrt[ 8 - Sqrt[Int3] (Sqrt[Int5] - 1) - Sqrt[10 + 2 Sqrt[Int5]] ] / 48 5 sqrt dup 1 sub 3 sqrt mul exch 2 mul 10 add sqrt add sub sqrt 4 div
Sin[22½°] = Cos[67½°] ≈ 0.382683432365\frac{1}{2} \sqrt{2 - \sqrt{2}}=Sqrt(2-Sqrt(2))/2Sqrt[2-Sqrt[Int2]]/22 2 sqrt sub sqrt 2 div
Sin[24°] = Cos[66°] ≈ 0.406736643076\frac{1}{8} \left(\sqrt{3} \left(1 + \sqrt{5}\right) - \sqrt{10 - 2 \sqrt{5}}\right)=( Sqrt(3)*(1+Sqrt(5)) - Sqrt(10-2*Sqrt(5)) ) / 8( Sqrt[Int3] (1+Sqrt[Int5]) - Sqrt[10-2 Sqrt[Int5]] ) / 85 sqrt dup 1 add 3 sqrt mul exch -2 mul 10 add sqrt sub 8 div
Sin[27°] = Cos[63°] ≈ 0.45399049974\frac{\sqrt{2} }{8}\left(1 + \sqrt{10 + 2 \sqrt{5}} - \sqrt{5}\right)=Sqrt(2) * ( 1 + Sqrt(10+2*Sqrt(5)) - Sqrt(5) ) / 8Sqrt[Int2] ( 1 + Sqrt[10+2 Sqrt[Int5]] - Sqrt[Int5] ) / 85 sqrt dup 2 mul 10 add sqrt exch sub 1 add 2 sqrt mul 8 div
Sin[28⅛°] = Cos[61⅞°] ≈ 0.471396736826\frac{1}{2} \sqrt{2 - \sqrt{2 - \sqrt{2 - \sqrt{2}}}}=Sqrt( 2 - Sqrt( 2 - Sqrt(2-Sqrt(2)) ) ) / 2Sqrt[ 2 - Sqrt[ 2 - Sqrt[2-Sqrt[Int2]] ] ] / 22 2 2 2 sqrt sub sqrt sub sqrt sub sqrt 2 div
Sin[30°] = Cos[60°] = 0.5\frac{1}{2}=1/21/21 2 div
Sin[33°] = Cos[57°] ≈ 0.544639035015\frac{1}{16} \left(2 \left(\sqrt{3} - 1\right) \sqrt{\sqrt{5} + 5} + \sqrt{2} \left(\sqrt{3} + 1\right) \left(\sqrt{5} - 1\right)\right)=( 2*(Sqrt(3)-1)*Sqrt(Sqrt(5)+5) + Sqrt(2)*(Sqrt(3)+1)*(Sqrt(5)-1) ) / 16( 2 (Sqrt[Int3]-1) Sqrt[Sqrt[Int5]+5] + Sqrt[Int2] (Sqrt[Int3]+1) (Sqrt[Int5]-1) ) / 163 sqrt 5 sqrt 2 copy 5 add sqrt exch 1 sub mul 2 mul 3 1 roll 1 sub exch 1 add mul 2 sqrt mul add 16 div
\frac{1}{4} \sqrt{8 + \sqrt{10 - 2 \sqrt{5}} - \sqrt{3} \left(1 + \sqrt{5}\right)}=Sqrt( 8 + Sqrt(10-2*Sqrt(5)) - Sqrt(3)*(1+Sqrt(5)) ) / 4Sqrt[ 8 + Sqrt[10-2 Sqrt[Int5]] - Sqrt[Int3] (1+Sqrt[Int5]) ] / 45 sqrt dup -2 mul 10 add sqrt 8 add exch 1 add 3 sqrt mul sub sqrt 4 div
Sin[33¾°] = Cos[56¼°] ≈ 0.55557023302\frac{1}{2} \sqrt{2 - \sqrt{2 - \sqrt{2}}}=Sqrt( 2 - Sqrt(2-Sqrt(2)) ) / 2Sqrt[ 2 - Sqrt[2-Sqrt[Int2]] ] / 22 2 2 sqrt sub sqrt sub sqrt 2 div
Sin[36°] = Cos[54°] ≈ 0.587785252292\sqrt{\frac{1}{8}\left(5 - \sqrt{5}\right)}=Sqrt((5-Sqrt(5))/8)Sqrt[(5-Sqrt[Int5])/8]5 5 sqrt sub 8 div sqrt
Sin[39°] = Cos[51°] ≈ 0.62932039105\frac{\sqrt{2} }{16}\left(\left(\sqrt{3} + 1\right) \left(\sqrt{5} + 1\right) - \left(\sqrt{3} - 1\right) \sqrt{10 - 2 \sqrt{5}}\right)=Sqrt(2) * ( (Sqrt(3)+1)*(Sqrt(5)+1) - (Sqrt(3)-1)*Sqrt(10-2*Sqrt(5)) ) / 16Sqrt[Int2] ( (Sqrt[Int3]+1) (Sqrt[Int5]+1) - (Sqrt[Int3]-1) Sqrt[10-2 Sqrt[Int5]] ) / 163 sqrt 5 sqrt 2 copy 1 add exch 1 add exch mul 3 1 roll -2 mul 10 add sqrt exch 1 sub mul sub 2 sqrt mul 16 div
\frac{1}{4} \sqrt{8 - \sqrt{10 + 2 \sqrt{5}} + \sqrt{3} \left(\sqrt{5} - 1\right)}=Sqrt( 8 - Sqrt(10+2*Sqrt(5)) + Sqrt(3)*(Sqrt(5)-1) ) / 4Sqrt[ 8 - Sqrt[10+2 Sqrt[Int5]] + Sqrt[Int3] (Sqrt[Int5]-1) ] / 45 sqrt dup 1 sub 3 sqrt mul 8 add exch 2 mul 10 add sqrt sub sqrt 4 div
Sin[39⅜°] = Cos[50⅝°] ≈ 0.634393284164\frac{1}{2} \sqrt{2 - \sqrt{2 - \sqrt{2 + \sqrt{2}}}}=Sqrt( 2 - Sqrt( 2 - Sqrt(2+Sqrt(2)) ) ) / 2Sqrt[ 2 - Sqrt[ 2 - Sqrt[2+Sqrt[Int2]] ] ] / 22 2 2 sqrt 2 add sqrt sub sqrt sub sqrt 2 div
Sin[42°] = Cos[48°] ≈ 0.669130606359\frac{1}{8} \left(1 + \sqrt{30 + 6 \sqrt{5}} - \sqrt{5}\right)=( 1 + Sqrt(30+6*Sqrt(5)) - Sqrt(5) ) / 8( 1 + Sqrt[30+6 Sqrt[Int5]] - Sqrt[Int5] ) / 85 sqrt dup 6 mul 30 add sqrt exch sub 1 add 8 div
Sin[45°] = Cos[45°] ≈ 0.707106781187\frac{\sqrt{2}}{2}=Sqrt(2)/2Sqrt[Int2]/22 sqrt 2 div
Sin[48°] = Cos[42°] ≈ 0.743144825477\frac{1}{8} \left(\sqrt{10 + 2 \sqrt{5}} + \sqrt{3} \left(\sqrt{5} - 1\right)\right)=( Sqrt(10+2*Sqrt(5)) + Sqrt(3)*(Sqrt(5)-1) ) / 8( Sqrt[10+2 Sqrt[Int5]] + Sqrt[Int3] (Sqrt[Int5]-1) ) / 85 sqrt dup 2 mul 10 add sqrt exch 1 sub 3 sqrt mul add 8 div
Sin[50⅝°] = Cos[39⅜°] ≈ 0.773010453363\frac{1}{2} \sqrt{2 + \sqrt{2 - \sqrt{2 + \sqrt{2}}}}=Sqrt( 2 + Sqrt( 2 - Sqrt(2+Sqrt(2)) ) ) / 2Sqrt[ 2 + Sqrt[ 2 - Sqrt[2+Sqrt[Int2]] ] ] / 22 2 sqrt 2 add sqrt sub sqrt 2 add sqrt 2 div
Sin[51°] = Cos[39°] ≈ 0.777145961457\frac{\sqrt{2} }{16}\left(\left(\sqrt{3} + 1\right) \sqrt{10 - 2 \sqrt{5}} + \left(\sqrt{3} - 1\right) \left(\sqrt{5} + 1\right)\right)=Sqrt(2) * ( (Sqrt(3)+1)*Sqrt(10-2*Sqrt(5)) + (Sqrt(3)-1)*(Sqrt(5)+1) ) / 16Sqrt[Int2] ( (Sqrt[Int3]+1) Sqrt[10-2 Sqrt[Int5]] + (Sqrt[Int3]-1) (Sqrt[Int5]+1) ) / 163 sqrt 5 sqrt 2 copy 1 add exch 1 sub mul 3 1 roll -2 mul 10 add sqrt exch 1 add mul add 2 sqrt mul 16 div
\frac{1}{4} \sqrt{8 + \sqrt{10 + 2 \sqrt{5}} - \sqrt{3} \left(\sqrt{5} - 1\right)}=Sqrt( 8 + Sqrt(10+2*Sqrt(5)) - Sqrt(3)*(Sqrt(5)-1) ) / 4Sqrt[ 8 + Sqrt[10+2 Sqrt[Int5]] - Sqrt[Int3] (Sqrt[Int5]-1) ] / 45 sqrt dup 2 mul 10 add sqrt 8 add exch 1 sub 3 sqrt mul sub sqrt 4 div
Sin[54°] = Cos[36°] ≈ 0.809016994375\frac{1}{4} \left(1 + \sqrt{5}\right)=(1+Sqrt(5))/4(1+Sqrt[Int5])/45 sqrt 1 add 4 div
Sin[56¼°] = Cos[33¾°] ≈ 0.831469612303\frac{1}{2} \sqrt{2 + \sqrt{2 - \sqrt{2}}}=Sqrt( 2 + Sqrt(2-Sqrt(2)) ) / 2Sqrt[ 2 + Sqrt[2-Sqrt[Int2]] ] / 22 2 sqrt sub sqrt 2 add sqrt 2 div
Sin[57°] = Cos[33°] ≈ 0.838670567945\frac{1}{16} \left(2 \left(\sqrt{3} + 1\right) \sqrt{\sqrt{5} + 5} - \sqrt{2} \left(\sqrt{3} - 1\right) \left(\sqrt{5} - 1\right)\right)=( 2*(Sqrt(3)+1)*Sqrt(Sqrt(5)+5) - Sqrt(2)*(Sqrt(3)-1)*(Sqrt(5)-1) ) / 16( 2 (Sqrt[Int3]+1) Sqrt[Sqrt[Int5]+5] - Sqrt[Int2] (Sqrt[Int3]-1) (Sqrt[Int5]-1) ) / 163 sqrt 5 sqrt 2 copy 5 add sqrt exch 1 add mul 2 mul 3 1 roll 1 sub exch 1 sub mul 2 sqrt mul sub 16 div
\frac{1}{4} \sqrt{8 - \sqrt{10 - 2 \sqrt{5}} + \sqrt{3} \left(1 + \sqrt{5}\right)}=Sqrt( 8 - Sqrt(10-2*Sqrt(5)) + Sqrt(3)*(1+Sqrt(5)) ) / 4Sqrt[ 8 - Sqrt[10-2 Sqrt[Int5]] + Sqrt[Int3] (1+Sqrt[Int5]) ] / 45 sqrt dup 1 add 3 sqrt mul exch -2 mul 10 add sqrt sub 8 add sqrt 4 div
Sin[60°] = Cos[30°] ≈ 0.866025403784\frac{\sqrt{3}}{2}=Sqrt(3)/2Sqrt[Int3]/23 sqrt 2 div
Sin[61⅞°] = Cos[28⅛°] ≈ 0.881921264348\frac{1}{2} \sqrt{2 + \sqrt{2 - \sqrt{2 - \sqrt{2}}}}=Sqrt( 2 + Sqrt( 2 - Sqrt(2-Sqrt(2)) ) ) / 2Sqrt[ 2 + Sqrt[ 2 - Sqrt[2-Sqrt[Int2]] ] ] / 22 2 2 sqrt sub sqrt sub sqrt 2 add sqrt 2 div
Sin[63°] = Cos[27°] ≈ 0.891006524188\frac{1}{8} \left(\sqrt{2} \left(\sqrt{5} - 1\right) + 2 \sqrt{\sqrt{5} + 5}\right)=( Sqrt(2)*(Sqrt(5)-1) + 2*Sqrt(Sqrt(5)+5) ) / 8( Sqrt[Int2] (Sqrt[Int5]-1) + 2 Sqrt[Sqrt[Int5]+5] ) / 85 sqrt dup 1 sub 2 sqrt mul exch 5 add sqrt 2 mul add 8 div
Sin[66°] = Cos[24°] ≈ 0.913545457643\frac{1}{8} \left(1 + \sqrt{30 - 6 \sqrt{5}} + \sqrt{5}\right)=( 1 + Sqrt(30-6*Sqrt(5)) + Sqrt(5) ) / 8( 1 + Sqrt[30-6 Sqrt[Int5]] + Sqrt[Int5] ) / 85 sqrt dup -6 mul 30 add sqrt add 1 add 8 div
Sin[67½°] = Cos[22½°] ≈ 0.923879532511\frac{1}{2} \sqrt{2 + \sqrt{2}}=Sqrt(2+Sqrt(2))/2Sqrt[2+Sqrt[Int2]]/22 sqrt 2 add sqrt 2 div
Sin[69°] = Cos[21°] ≈ 0.933580426497\frac{1}{16} \left(2 \left(\sqrt{3} - 1\right) \sqrt{5 - \sqrt{5}} + \sqrt{2} \left(\sqrt{3} + 1\right) \left(\sqrt{5} + 1\right)\right)=( 2*(Sqrt(3)-1)*Sqrt(5-Sqrt(5)) + Sqrt(2)*(Sqrt(3)+1)*(Sqrt(5)+1) ) / 16( 2 (Sqrt[Int3]-1) Sqrt[5-Sqrt[Int5]] + Sqrt[Int2] (Sqrt[Int3]+1) (Sqrt[Int5]+1) ) / 163 sqrt 5 sqrt 2 copy 5 exch sub sqrt exch 1 sub mul 2 mul 3 1 roll 1 add exch 1 add mul 2 sqrt mul add 16 div
\frac{1}{4} \sqrt{8 + \sqrt{3} \left(\sqrt{5} - 1\right) + \sqrt{10 + 2 \sqrt{5}}}=Sqrt( 8 + Sqrt(3)*(Sqrt(5)-1) + Sqrt(10+2*Sqrt(5)) ) / 4Sqrt[ 8 + Sqrt[Int3] (Sqrt[Int5]-1) + Sqrt[10+2 Sqrt[Int5]] ] / 45 sqrt dup 2 mul 10 add sqrt 8 add exch 1 sub 3 sqrt mul add sqrt 4 div
Sin[72°] = Cos[18°] ≈ 0.951056516295\sqrt{\frac{1}{8}\left(5 + \sqrt{5}\right)}=Sqrt((5+Sqrt(5))/8)Sqrt[(5+Sqrt[Int5])/8]5 sqrt 5 add 8 div sqrt
Sin[73⅛°] = Cos[16⅞°] ≈ 0.956940335732\frac{1}{2} \sqrt{2 + \sqrt{2 + \sqrt{2 - \sqrt{2}}}}=Sqrt( 2 + Sqrt( 2 + Sqrt(2-Sqrt(2)) ) ) / 2Sqrt[ 2 + Sqrt[ 2 + Sqrt[2-Sqrt[Int2]] ] ] / 22 2 sqrt sub sqrt 2 add sqrt 2 add sqrt 2 div
Sin[75°] = Cos[15°] ≈ 0.965925826289\frac{\sqrt{2} }{4}\left(\sqrt{3} + 1\right)=Sqrt(2) * (Sqrt(3)+1) / 4Sqrt[Int2] (Sqrt[Int3]+1) / 43 sqrt 1 add 2 sqrt mul 4 div
Sin[78°] = Cos[12°] ≈ 0.978147600734\frac{1}{8} \left(\sqrt{30 + 6 \sqrt{5}} + \sqrt{5} - 1\right)=( Sqrt(30+6*Sqrt(5)) + Sqrt(5) - 1 ) / 8( Sqrt[30+6 Sqrt[Int5]] + Sqrt[Int5] - 1 ) / 85 sqrt dup 6 mul 30 add sqrt add 1 sub 8 div
Sin[78¾°] = Cos[11¼°] ≈ 0.980785280403\frac{1}{2} \sqrt{2 + \sqrt{2 + \sqrt{2}}}=Sqrt( 2 + Sqrt(2+Sqrt(2)) ) / 2Sqrt[ 2 + Sqrt[2+Sqrt[Int2]] ] / 22 sqrt 2 add sqrt 2 add sqrt 2 div
Sin[81°] = Cos[9°] ≈ 0.987688340595\frac{1}{8} \left(2 \sqrt{5 - \sqrt{5}} + \sqrt{2} \left(\sqrt{5} + 1\right)\right)=( 2*Sqrt(5-Sqrt(5)) + Sqrt(2)*(Sqrt(5)+1) ) / 8( 2 Sqrt[5-Sqrt[Int5]] + Sqrt[Int2] (Sqrt[Int5]+1) ) / 85 sqrt dup 1 add 2 sqrt mul exch 5 exch sub sqrt 2 mul add 8 div
Sin[84°] = Cos[6°] ≈ 0.994521895368\frac{1}{8} \left(\sqrt{10 - 2 \sqrt{5}} + \sqrt{3} \left(1 + \sqrt{5}\right)\right)=( Sqrt(10-2*Sqrt(5)) + Sqrt(3)*(1+Sqrt(5)) ) / 8( Sqrt[10-2 Sqrt[Int5]] + Sqrt[Int3] (1+Sqrt[Int5]) ) / 85 sqrt dup 1 add 3 sqrt mul exch -2 mul 10 add sqrt add 8 div
Sin[84⅜°] = Cos[5⅝°] ≈ 0.995184726672\frac{1}{2} \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}=Sqrt( 2 + Sqrt( 2 + Sqrt(2+Sqrt(2)) ) ) / 2Sqrt[ 2 + Sqrt[ 2 + Sqrt[2+Sqrt[Int2]] ] ] / 22 sqrt 2 add sqrt 2 add sqrt 2 add sqrt 2 div
Sin[87°] = Cos[3°] ≈ 0.998629534755\frac{1}{16} \left(\sqrt{2} \left(\sqrt{3} - 1\right) \left(\sqrt{5} - 1\right) + 2 \left(\sqrt{3} + 1\right) \sqrt{\sqrt{5} + 5}\right)=( Sqrt(2)*(Sqrt(3)-1)*(Sqrt(5)-1) + 2*(Sqrt(3)+1)*Sqrt(Sqrt(5)+5) ) / 16( Sqrt[Int2] (Sqrt[Int3]-1) (Sqrt[Int5]-1) + 2 (Sqrt[Int3]+1) Sqrt[Sqrt[Int5]+5] ) / 163 sqrt 5 sqrt 2 copy 5 add sqrt exch 1 add mul 2 mul 3 1 roll 1 sub exch 1 sub mul 2 sqrt mul add 16 div
\frac{1}{4} \sqrt{8 + \sqrt{10 - 2 \sqrt{5}} + \sqrt{3} \left(1 + \sqrt{5}\right)}=Sqrt( 8 + Sqrt(10-2*Sqrt(5)) + Sqrt(3)*(1+Sqrt(5)) ) / 4Sqrt[ 8 + Sqrt[10-2 Sqrt[Int5]] + Sqrt[Int3] (1+Sqrt[Int5]) ] / 45 sqrt dup 1 add 3 sqrt mul 8 add exch -2 mul 10 add sqrt add sqrt 4 div
Sin[90°] = Cos[0°] = 11111

Julian D. A. Wiseman, June 2008