Question

Write the principale value of cos inverse cos( 2π/3)+sin i verse of sine 2π/3

Answer 1

cos-1(cos2π/3) + sin-1(sin2π/3) =

as we know cos-1(cosx) = x for R ∈ [0, π] and sin-1(sinx) = x for R ∈ [-π/2,π/2]

so cos-1(cos2π/3) = 2π/3      -------(i)

 and sin-1(sin2π/3) is not 2π/3 as 2π/3 do not belong to [-π/2,π/2]

so (sin2π/3) = sin (π-2π/3) = sin( π/3)

and π/3 belong to [-π/2,π/2]

so sin-1(sinπ/3) = π/3    ---------(ii) 

adding (i) and (ii) we get

⇒ 2π/3 + π/3 = π