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4.Cálculo de las funciones seno y coseno de los ángulos de 24o,12o,6o,3o

Tenemos que 54o - 30o = 24o, así:

sen24o = sen$\displaystyle \left(\vphantom{54^o-30^o}\right.$54o - 30o$\displaystyle \left.\vphantom{54^o-30^o}\right)$
  = sen 54o cos 30o - cos 54o sen 30o
  = $\displaystyle {1\over 8}$$\displaystyle \left[\vphantom{\sqrt{15}+\sqrt{3}-\sqrt{10-2\sqrt{5}}}\right.$$\displaystyle \sqrt{15}$ + $\displaystyle \sqrt{3}$ - $\displaystyle \sqrt{10-2\sqrt{5}}$$\displaystyle \left.\vphantom{\sqrt{15}+\sqrt{3}-\sqrt{10-2\sqrt{5}}}\right]$
     
cos 24o = cos 54o cos 30o + sen 54o sen 30o
  = $\displaystyle {1\over 8}$$\displaystyle \left[\vphantom{\sqrt{5}+ 1 -\sqrt{10-6\sqrt{5}}}\right.$$\displaystyle \sqrt{5}$ + 1 - $\displaystyle \sqrt{10-6\sqrt{5}}$$\displaystyle \left.\vphantom{\sqrt{5}+ 1 -\sqrt{10-6\sqrt{5}}}\right]$
     
sen 12o = $\displaystyle {sen 24^o\over 2}$
  = $\displaystyle {\sqrt{1-cos 24^o\over 2}}$
  = $\displaystyle {\sqrt{7-\sqrt{5}-\sqrt{30-6\sqrt{5}}}\over 4}$
     
cos12o = $\displaystyle {cos 24^o\over 2}$
  = $\displaystyle {\sqrt{9+\sqrt{5}+\sqrt{30-6\sqrt{5}}}\over 4}$

 

También se tiene que 6o = 36o - 30o y entonces :

sen 6o = sen 36o cos 30o - cos 36o sen 30o
  = $\displaystyle {1\over 8}$$\displaystyle \left[\vphantom{\sqrt{30-6\sqrt{5}}-\sqrt{5}-1}\right.$$\displaystyle \sqrt{30-6\sqrt{5}}$ - $\displaystyle \sqrt{5}$ - 1$\displaystyle \left.\vphantom{\sqrt{30-6\sqrt{5}}-\sqrt{5}-1}\right]$
     
cos 6o = cos 36o cos 30o - sen 36o sen 30o
  = $\displaystyle {1\over 8}$$\displaystyle \left[\vphantom{\sqrt{10-2\sqrt{5}}+\sqrt{15}+\sqrt{3}}\right.$$\displaystyle \sqrt{10-2\sqrt{5}}$ + $\displaystyle \sqrt{15}$ + $\displaystyle \sqrt{3}$$\displaystyle \left.\vphantom{\sqrt{10-2\sqrt{5}}+\sqrt{15}+\sqrt{3}}\right]$

 

Con las fórmulas del ángulo medio se obtiene:

sen 3o = $\displaystyle {\sqrt{8-\sqrt{15}-\sqrt{3}-\sqrt{10}-2\sqrt{5}}\over 4}$
     
cos 3o = $\displaystyle {\sqrt{8+\sqrt{15}+\sqrt{3}+\sqrt{10}+2\sqrt{5}}\over 4}$

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