Question

СРОЧНО!!! 15 БАЛЛОВ!!!

tg ( arcsin1/3 + arccos2/3) = ?

Answer 1

Обозначим
arcsin 1/3 =α⇒    sin α=1/3 , α∈[-π/2;π/2]
cosα=√(1-sin²α)=√(1-(1/9))=√(8/9)=2√2/3
tgα=1/(2√2)

Обозначим
arccos 2/3 =β⇒     cos β=2/3 , β∈[0;π]
sinβ=√(1-cos²β)=√(1-(4/9))=√(5/9)=√5/3
tgβ=√5

$tg(arcsin frac{1}{3}+arccos frac{2}{3})=tg( alpha + beta )= frac{tg alpha +tg beta }{1-tg alphacdot tg beta } = \ = frac {frac{1}{2 sqrt{2} }+ sqrt{5}}{1- frac{1}{2 sqrt{2}cdot sqrt{5} } } = frac{1+2 sqrt{10} }{ frac{2 sqrt{10}-1 }{ sqrt{5} } } = frac{ sqrt{5}(2 sqrt{10}+1) }{2 sqrt{10}-1 }= frac{ sqrt{5}(2 sqrt{10}+1)(2 sqrt{10}+1) }{(2 sqrt{10}-1)(2 sqrt{10}+1) }= \ = frac{41 sqrt{5}+20 sqrt{2}}{39}$