Find the principal value of:() sin" ((i) cos" (MB

1.	Find the principal value of:() sin" ((i) cos" (MB
Answer» ii. is the correct answer (i) If θ be the principal value of sin^−1 x then - π/2 ≤ θ ≤ π/2. Therefore, If the principal value of sin^−1(-1/2) be θ then sin^−1(-1/2) = θ ⇒ sin θ = - 1/2 = sin (-π/6) [Since, - π/2 ≤ θ ≤ π/2] Therefore, the principal value of sin^−1(-1/2) is (-π/6). (ii) If the principal value of cos^−1 x is θ then we know, 0 ≤ θ ≤ π. Therefore, If the principal value of cos^−1(- √3/2) be θ then cos^−1(- √3/2) = θ ⇒ cos θ = (- √3/2) = cos π/6 = cos (π - π/6) [Since, 0 ≤ θ ≤ π] Therefore, the principal value of cos^−1(- √3/2) is π - π/6 = 5π/6.

Answer»

ii. is the correct answer

(i) If θ be the principal value of sin^−1 x then - π/2 ≤ θ ≤ π/2.

Therefore, If the principal value of sin^−1(-1/2) be θ then sin^−1(-1/2) = θ

⇒ sin θ = - 1/2 = sin (-π/6) [Since, - π/2 ≤ θ ≤ π/2]

Therefore, the principal value of sin^−1(-1/2) is (-π/6).

(ii) If the principal value of cos^−1 x is θ then we know, 0 ≤ θ ≤ π.

Therefore, If the principal value of cos^−1(- √3/2) be θ then cos^−1(- √3/2) = θ

⇒ cos θ = (- √3/2) = cos π/6 = cos (π - π/6) [Since, 0 ≤ θ ≤ π]

Therefore, the principal value of cos^−1(- √3/2) is π - π/6 = 5π/6.

Discussion