Найдите ctg a/2 если sina=1/3 90"a"180

найдите ctg a/2 если sina=1/3 90"a"180

ctg² (a/2) = 1 / sin² (a/2) - 1 = cos² (a/2) / sin² (a/2) = [1+cos (a) ] / [1-cos (a) ] =

(cos (a) = - √ (1-sin² (a), т. к. a∉ (90; 180))

=[1-√ (1-sin² (a) ]/[1+√ (1-sin² (a) ]=[1-√ (1-sin² (a) ]² / [1 - (1-sin² (a) ]=

=[1-√ (1 - (1/3) ²]² / [ (1/3)) ]=9[1-√ (8/9) ]²

т. о ctg² (a/2) = 9[1-√ (8/9) ]² ⇒ctg (a/2) = 3[1-√ (8/9) ] = 3[3-√ (8) ]/3=3-2 √2

(a/2∈ (90°; 45°) поэтому ctg (a/2) >0)