Вычислите: cos (arctg (-3/4) + arcctg (-1/√3))

1. Выразим функции синус и косинус через функции тангенс и котангенс:

a) cosx;

sin^2 (x) + cos^2 (x) = 1; tg^2 (x) + 1 = 1/cos^2 (x); cos^2 (x) = 1 / (1 + tg^2 (x)); cosx = ± 1/√ (1 + tg^2 (x)); (1) cosx = ± 1/√ (1 + 1/ctg^2 (x)). (2)

b) sinx;

sin^2 (x) + cos^2 (x) = 1; 1 + ctg^2 (x) = 1/sin^2 (x); sin^2 (x) = 1 / (1 + ctg^2 (x)); sinx = ± 1/√ (1 + ctg^2 (x)); (3) sinx = ± 1/√ (1 + 1/tg^2 (x)). (4)

2. Обозначим заданное выражение Z и по формуле для косинуса суммы двух углов преобразуем его:

cos (a + b) = cosa * cosb - sina * sinb; Z = cos (arctg (-3/4) + arcctg (-1/√3)); Z = cos (-arctg (3/4) - arcctg (1/√3)); Z = cos (arctg (3/4) + arcctg (1/√3)); Z = cos (arctg (3/4)) * cos (arcctg (1/√3)) - sin (arctg (3/4)) * sin (arcctg (1/√3)); cos (arctg (3/4)) = 1/√ (1 + (3/4) ^2) = 1/√ (1 + 9/16) = 1/√ (25/16) = 4/5; sin (arctg (3/4)) = 3/5; cos (arctg (1/√3)) = 1/√ (1 + (1/√3) ^2) = 1/√ (1 + 1/3) = 1/√ (4/3) = √3/2; sin (arctg (1/√3)) = 1/2; Z = 4/5 * √3/2 - 3/5 * 1/2; Z = 4√3/10 - 3/10 = (4√3 - 3) / 10.

Ответ: (4√3 - 3) / 10.