Trigonometric Table: multiples of 3 degrees mostly in radicals

Sines and cosines exists for all multiples of 3 degrees in radicals. Tangent and cotangent may in some case also; but, in some cases, roots of \(8^\text{th}\) order polynomials are required.

[[The trigonometric table from 3 to 87 by 3 degrees|Section Heading]]

\[ \begin{array}{lllllll} 3 & \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) & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+2} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,5,0\right] & \frac{16}{\sqrt{2} \left(\sqrt{3}+1\right) \left(\sqrt{5}-1\right)-2 \left(\sqrt{3}-1\right) \sqrt{\sqrt{5}+5}} & 2 \sqrt{\frac{2}{\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4}} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,8,0\right] \\ 6 & \frac{1}{8} \left(-\sqrt{5}+\sqrt{30-6 \sqrt{5}}-1\right) & \frac{1}{4} \sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7} & \sqrt{-2 \sqrt{5}-2 \sqrt{15-6 \sqrt{5}}+7} & \sqrt{5}+\sqrt{6 \sqrt{5}+15}+2 & \frac{4}{\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & \sqrt{10 \sqrt{5}+2 \sqrt{3 \left(38 \sqrt{5}+85\right)}+23} \\ 9 & \frac{1}{4} \sqrt{8-2 \sqrt{2 \left(\sqrt{5}+5\right)}} & \frac{1}{2} \sqrt{\sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)}+2} & \sqrt{5}-\sqrt{2 \sqrt{5}+5}+1 & \frac{4}{\sqrt{8-2 \sqrt{2 \left(\sqrt{5}+5\right)}}} & \frac{2}{\sqrt{\sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)}+2}} & \sqrt{5}+\sqrt{2 \sqrt{5}+5}+1 \\ 12 & \frac{1}{4} \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7} & \frac{1}{8} \left(\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}-1\right) & \sqrt{-10 \sqrt{5}-2 \sqrt{3 \left(85-38 \sqrt{5}\right)}+23} & \sqrt{2 \left(\sqrt{5}+\sqrt{6 \sqrt{5}+15}+4\right)} & -\sqrt{5}+\sqrt{15-6 \sqrt{5}}+2 & \sqrt{2 \sqrt{5}+2 \sqrt{6 \sqrt{5}+15}+7} \\ 15 & \frac{\sqrt{2-\sqrt{3}}}{2} & \frac{\sqrt{\sqrt{3}+2}}{2} & 2-\sqrt{3} & 2 \sqrt{\sqrt{3}+2} & \frac{2}{\sqrt{\sqrt{3}+2}} & \sqrt{3}+2 \\ 18 & \frac{1}{4} \left(\sqrt{5}-1\right) & \frac{1}{2} \sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)} & \sqrt{1-\frac{2}{\sqrt{5}}} & \sqrt{5}+1 & \sqrt{2-\frac{2}{\sqrt{5}}} & \sqrt{2 \sqrt{5}+5} \\ 21 & \frac{1}{2 \sqrt{-\frac{2}{\sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7}-4}}} & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{-\sqrt{5}+\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,5,0\right] & 2 \sqrt{-\frac{2}{\sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7}-4}} & \frac{2}{\sqrt{\frac{1}{2} \sqrt{-\sqrt{5}+\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,8,0\right] \\ 24 & \frac{1}{4} \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7} & \frac{1}{8} \left(\sqrt{5}+\sqrt{30-6 \sqrt{5}}+1\right) & \sqrt{10 \sqrt{5}-2 \sqrt{3 \left(38 \sqrt{5}+85\right)}+23} & \frac{4}{\sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & -\sqrt{5}+\sqrt{6 \sqrt{5}+15}-2 & \sqrt{-2 \sqrt{5}+2 \sqrt{15-6 \sqrt{5}}+7} \\ 27 & \frac{1}{2} \sqrt{2-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}} & \frac{1}{2} \sqrt{\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}+2} & \sqrt{5}-\sqrt{5-2 \sqrt{5}}-1 & \frac{2}{\sqrt{2-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}}} & \frac{2}{\sqrt{\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}+2}} & \sqrt{5}+\sqrt{5-2 \sqrt{5}}-1 \\ 30 & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{3}} & 2 & \frac{2}{\sqrt{3}} & \sqrt{3} \\ 33 & \frac{1}{4} \sqrt{8-2 \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}+2} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,6,0\right] & \frac{4}{\sqrt{8-2 \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}}} & 2 \sqrt{\frac{2}{\sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,7,0\right] \\ 36 & \frac{1}{4} \sqrt{10-2 \sqrt{5}} & \frac{1}{4} \left(\sqrt{5}+1\right) & \sqrt{5-2 \sqrt{5}} & \sqrt{2+\frac{2}{\sqrt{5}}} & \sqrt{5}-1 & \sqrt{1+\frac{2}{\sqrt{5}}} \\ 39 & \frac{1}{4} \sqrt{8-2 \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7}} & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{-\sqrt{5}-\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,6,0\right] & \frac{4}{\sqrt{8-2 \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7}}} & \frac{2}{\sqrt{\frac{1}{2} \sqrt{-\sqrt{5}-\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2}} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,7,0\right] \\ 42 & \frac{1}{8} \left(-\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+1\right) & \frac{1}{4} \sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7} & \sqrt{2 \sqrt{5}-2 \sqrt{6 \sqrt{5}+15}+7} & \sqrt{5}+\sqrt{15-6 \sqrt{5}}-2 & \sqrt{2 \left(\sqrt{5}-\sqrt{6 \sqrt{5}+15}+4\right)} & \sqrt{-10 \sqrt{5}+2 \sqrt{3 \left(85-38 \sqrt{5}\right)}+23} \\ 45 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 & \sqrt{2} & \sqrt{2} & 1 \\ 48 & \frac{1}{4} \sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7} & \frac{1}{8} \left(-\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+1\right) & \sqrt{-10 \sqrt{5}+2 \sqrt{3 \left(85-38 \sqrt{5}\right)}+23} & \sqrt{2 \left(\sqrt{5}-\sqrt{6 \sqrt{5}+15}+4\right)} & \sqrt{5}+\sqrt{15-6 \sqrt{5}}-2 & \sqrt{2 \sqrt{5}-2 \sqrt{6 \sqrt{5}+15}+7} \\ 51 & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{-\sqrt{5}-\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2} & \frac{1}{4} \sqrt{8-2 \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7}} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,7,0\right] & \frac{2}{\sqrt{\frac{1}{2} \sqrt{-\sqrt{5}-\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2}} & \frac{4}{\sqrt{8-2 \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7}}} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,6,0\right] \\ 54 & \frac{1}{4} \left(\sqrt{5}+1\right) & \frac{1}{4} \sqrt{10-2 \sqrt{5}} & \sqrt{1+\frac{2}{\sqrt{5}}} & \sqrt{5}-1 & \sqrt{2+\frac{2}{\sqrt{5}}} & \sqrt{5-2 \sqrt{5}} \\ 57 & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}+2} & \frac{1}{4} \sqrt{8-2 \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,7,0\right] & 2 \sqrt{\frac{2}{\sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4}} & \frac{4}{\sqrt{8-2 \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,6,0\right] \\ 60 & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} & \frac{2}{\sqrt{3}} & 2 & \frac{1}{\sqrt{3}} \\ 63 & \frac{1}{2} \sqrt{\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}+2} & \frac{1}{2} \sqrt{2-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}} & \sqrt{5}+\sqrt{5-2 \sqrt{5}}-1 & \frac{2}{\sqrt{\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}+2}} & \frac{2}{\sqrt{2-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)}}} & \sqrt{5}-\sqrt{5-2 \sqrt{5}}-1 \\ 66 & \frac{1}{8} \left(\sqrt{5}+\sqrt{30-6 \sqrt{5}}+1\right) & \frac{1}{4} \sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7} & \sqrt{-2 \sqrt{5}+2 \sqrt{15-6 \sqrt{5}}+7} & -\sqrt{5}+\sqrt{6 \sqrt{5}+15}-2 & \frac{4}{\sqrt{\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & \sqrt{10 \sqrt{5}-2 \sqrt{3 \left(38 \sqrt{5}+85\right)}+23} \\ 69 & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{-\sqrt{5}+\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2} & \frac{1}{4} \sqrt{8-2 \sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,8,0\right] & \frac{2}{\sqrt{\frac{1}{2} \sqrt{-\sqrt{5}+\sqrt{6 \left(5-\sqrt{5}\right)}+7}+2}} & \frac{4}{\sqrt{8-2 \sqrt{-\sqrt{5}+\sqrt{30-6 \sqrt{5}}+7}}} & \text{Root}\left[\text{$\#$1}^8+16 \text{$\#$1}^7-60 \text{$\#$1}^6-16 \text{$\#$1}^5+134 \text{$\#$1}^4-16 \text{$\#$1}^3-60 \text{$\#$1}^2+16 \text{$\#$1}+1\&,5,0\right] \\ 72 & \frac{1}{2} \sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)} & \frac{1}{4} \left(\sqrt{5}-1\right) & \sqrt{2 \sqrt{5}+5} & \sqrt{2-\frac{2}{\sqrt{5}}} & \sqrt{5}+1 & \sqrt{1-\frac{2}{\sqrt{5}}} \\ 75 & \frac{\sqrt{\sqrt{3}+2}}{2} & \frac{\sqrt{2-\sqrt{3}}}{2} & \sqrt{3}+2 & \frac{2}{\sqrt{\sqrt{3}+2}} & 2 \sqrt{\sqrt{3}+2} & 2-\sqrt{3} \\ 78 & \frac{1}{8} \left(\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}-1\right) & \frac{1}{4} \sqrt{-\sqrt{5}-\sqrt{30-6 \sqrt{5}}+7} & \sqrt{2 \sqrt{5}+2 \sqrt{6 \sqrt{5}+15}+7} & -\sqrt{5}+\sqrt{15-6 \sqrt{5}}+2 & \sqrt{2 \left(\sqrt{5}+\sqrt{6 \sqrt{5}+15}+4\right)} & \sqrt{-10 \sqrt{5}-2 \sqrt{3 \left(85-38 \sqrt{5}\right)}+23} \\ 81 & \frac{1}{2} \sqrt{\sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)}+2} & \frac{1}{4} \sqrt{8-2 \sqrt{2 \left(\sqrt{5}+5\right)}} & \sqrt{5}+\sqrt{2 \sqrt{5}+5}+1 & \frac{2}{\sqrt{\sqrt{\frac{1}{2} \left(\sqrt{5}+5\right)}+2}} & \frac{4}{\sqrt{8-2 \sqrt{2 \left(\sqrt{5}+5\right)}}} & \sqrt{5}-\sqrt{2 \sqrt{5}+5}+1 \\ 84 & \frac{1}{4} \sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7} & \frac{1}{8} \left(-\sqrt{5}+\sqrt{30-6 \sqrt{5}}-1\right) & \sqrt{10 \sqrt{5}+2 \sqrt{3 \left(38 \sqrt{5}+85\right)}+23} & \frac{4}{\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}} & \sqrt{5}+\sqrt{6 \sqrt{5}+15}+2 & \sqrt{-2 \sqrt{5}-2 \sqrt{15-6 \sqrt{5}}+7} \\ 87 & \frac{1}{2} \sqrt{\frac{1}{2} \sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+2} & \frac{1}{2} \sqrt{\frac{-\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+9}{2 \left(\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4\right)}} & \text{Root}\left[\text{$\#$1}^8-16 \text{$\#$1}^7-60 \text{$\#$1}^6+16 \text{$\#$1}^5+134 \text{$\#$1}^4+16 \text{$\#$1}^3-60 \text{$\#$1}^2-16 \text{$\#$1}+1\&,8,0\right] & 2 \sqrt{\frac{2}{\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4}} & 2 \sqrt{\frac{2 \left(\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4\right)}{-\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+9}} & \frac{\sqrt{-\sqrt{5}-\sqrt{6 \left(\sqrt{5}+5\right)}+9}}{\sqrt{\sqrt{5}+\sqrt{6 \left(\sqrt{5}+5\right)}+7}+4} \\ \end{array} \]