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Asif Hossain
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Joined:Sat Jan 02, 2021 9:28 pm
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Unread post by Asif Hossain » Thu May 27, 2021 10:17 am

In geometry, the Conway triangle notation, named after John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically. Given a reference triangle whose sides are a, b and c and whose corresponding internal angles are A, B, and C then the Conway triangle notation is simply represented as follows: S = b c sin ⁡ A = a c sin ⁡ B = a b sin ⁡ C {\displaystyle S=bc\sin A=ac\sin B=ab\sin C\,} where S = 2 × area of reference triangle and S φ = S cot ⁡ φ. {\displaystyle S_{\varphi }=S\cot \varphi.\,} in particular S A = S cot ⁡ A = b c cos ⁡ A = b 2 + c 2 − a 2 {\displaystyle S_{A}=S\cot A=bc\cos A={\frac {b^{2}+c^{2}-a^{2}}{2}}\,} S B = S cot ⁡ B = a c cos ⁡ B = a 2 + c 2 − b 2 {\displaystyle S_{B}=S\cot B=ac\cos B={\frac {a^{2}+c^{2}-b^{2}}{2}}\,} S C = S cot ⁡ C = a b cos ⁡ C = a 2 + b 2 − c 2 {\displaystyle S_{C}=S\cot C=ab\cos C={\frac {a^{2}+b^{2}-c^{2}}{2}}\,} S ω = S cot ⁡ ω = a 2 + b 2 + c 2 {\displaystyle S_{\omega }=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}}\,} where ω {\displaystyle \omega \,} is the Brocard angle. The law of cosines is used: a 2 = b 2 + c 2 − 2 b c cos ⁡ A {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A}. S π 3 = S cot ⁡ π 3 = S 3 {\displaystyle S_{\frac {\pi }{3}}=S\cot {\frac {\pi }{3}}=S{\frac {\sqrt {3}}{3}}\,} S 2 φ = S φ 2 − S 2 S φ S φ 2 = S φ + S φ 2 + S 2 {\displaystyle S_{2\varphi }={\frac {S_{\varphi }^{2}-S^{2}}{2S_{\varphi }}}\\quad S_{\frac {\varphi }{2}}=S_{\varphi }+{\sqrt {S_{\varphi }^{2}+S^{2}}}\,} for values of φ {\displaystyle \varphi } where 0 < φ < π {\displaystyle 0
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Hmm..Hammer...Treat everything as nail

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