Write the value of cos-1(\(\frac 12\)) + 2 sin-1(\(\frac 12\)).

1.	Write the value of cos-1(\(\frac 12\)) + 2 sin-1(\(\frac 12\)).
Answer» Let cos^-1(1/2) = x and sin ^-1 (1/2) = y ⇒ cos x = 1/2 and sin y = 1/2 We know that the range of the principal value branch of sin^-1 is (-π/2, π/2) and cos^-1 is (0, π). We also know that sin (π/ 6) = 1/2 and cos (π/3) = 1/2 ⇒ Value of cos^-1(1/2) + 2sin^-1 (1/2) = π/3 + 2(π/6) = π/3 + π/3 = 2π/3 ∴ Value of cos^-1 (1/2) + 2sin^-1 (1/2) = 2π/3

Answer»

Let cos^-1(1/2) = x and sin ^-1 (1/2) = y

⇒ cos x = 1/2 and sin y = 1/2

We know that the range of the principal value branch of sin^-1 is (-π/2, π/2) and cos^-1 is (0, π).

We also know that sin (π/ 6) = 1/2 and cos (π/3) = 1/2

⇒ Value of cos^-1(1/2) + 2sin^-1 (1/2) = π/3 + 2(π/6)

= π/3 + π/3

= 2π/3

∴ Value of cos^-1 (1/2) + 2sin^-1 (1/2) = 2π/3

Discussion