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sin(α/2) – formula’s demonstration

Posted in Math Theory by Lia on 02/23/2013

For a list with trigonometric identities click  here

 

Let’s start with trigonometric identities:

[1]    tabular{11}{11}{{cos{2}{beta} = cos^2{beta} - sin^2{beta}}}  and

[2]    tabular{11}{11}{{ cos^2{beta} + sin^2{beta} = 1}}

 

Writing the expression in [2] in a different way we have:

[3]    tabular{11}{11}{{ cos^2{beta}  = 1 - sin^2 {beta}}}

By replacing  {cos^2{beta}} in [1] with the value in [3] we have:

cos{2}{beta} = 1 - sin^2{beta} - sin^2{beta}   {}{right}{}   cos{2}{beta}  = 1 - 2sin^2{beta}   {}{right}{}   2{sin^2}{beta}  = 1 - cos{2}{beta}

by replasing  in the last expression {}{beta}{}   with   {}{{alpha}/2}{}  we have:

 2{sin^2}({alpha}/2)  = 1 - cos{alpha}   {}{right}{}   {sin^2}({alpha}/2)  = {{1 - cos{alpha}}/2}

 

↓↓

 

tabular{11}{11}{{sin{{alpha}/2} = {pm}{sqrt{{1 - cos{alpha}}/2}}}}

 

 

 

 

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