Question

ДАМ 50 БАЛЛОВ ВЫЧИСЛИТЬ 3sin(arctg1/4+arcctg1/4)

Answer 1

Вспомним формулы
sin(x+y) = sin(x)cos(y) + sin(y)cos(x)
sin(arctg(x))= x/√(1+x²)
cos(arctg(x))=1/√(1+x²) 
sin(arcctg(x))=1/√(1+x²) 
cos(arcctg(x))=x/√(1+x²)
3sin(arctg1/4+arcctg1/4) = 3 ( sin(arctg(1/4)*cos(arcctg(1/4) + sin(arcctg(1/4)*cos(arctg(1/4)) = 3*( 1/4 / √(1+1/4²)*1/4/√(1+1/4²) + 1/√(1+1/4²)*1/√(1+1/4²)) = 3*(1/4²/(1+1/4²) + 1/(1+1/4²)) = 3*( (1+1/4²)/(1+1/4²)) =3*1=3
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а можно вспомнить два замечательных тождества
arcsin x + arccos x = π/2
acrtg x + arcctg x = π/2
3sin(arctg1/4+arcctg1/4) = 3 sin (π/2) = 3  (sin π/2 = 1)

Answer 2

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