Вычислить : sin (arccos (sqrt (2) / 3) + arctg (sqrt (5))

Sin (arccos (√2/3) + arctg√5)

Арккосинус √2/3 - это угол, α, косинус которого равен √2/3.

arccos (√2/3) = α α∈[0; π]

cos α = √2/3

arctg√5 = β, β∈[ - π/2; π/2]

tgβ = √5

sin (arccos (√2/3) + arctg√5) = sin (α + β) = sinα·cosβ + cosα·sinβ

sinα = √ (1 - cos²α) = √ (1 - 2/9) = √7/3

tg²β + 1 = 1/cos²β

5 + 1 = 1/cos²β

cos²β = 1/6

cosβ = 1/√6

sinβ = √ (1 - cos²β) = √ (1 - 1/6) = √ (5/6)

sinα·cosβ + cosα·sinβ = √7/3 · 1/√6 + √2/3 · √5/√6 =

= (√7 + √10) / (3√6)