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**Q.1. Find the principal value of sin ^{-1} (-1 / 2).**

sin y = -1 / 2 = -sin (Ï€ / 6) = sin (-Ï€ / 6)

We know that the range of the principal value branch of sin

Therefore, principal value of sin

cos y = âˆš3 / 2 = cos(Ï€ / 6)

We know that the range of the principal value branch of cos

Therefore, principal value of cos

We know that the range of the principal value branch of cosec

Therefore, principal value of cos

We know that the range of the principal value branch of tan

Therefore, principal value of tan

We know that the range of the principal value branch of cos

Therefore, principal value of cos

We know that the range of the principal value branch of tan

Therefore, principal value of tan

We know that the range of the principal value branch of sec

Therefore, principal value of sec

We know that the range of the principal value branch of cot

Therefore, principal value of cot

cos y = -1 / âˆš2 = -cos(Ï€ / 4) = cos(Ï€ - Ï€ / 4) = cos (3Ï€ / 4)

We know that the range of the principal value branch of cos

Therefore, principal value of cos

cosec y = -âˆš2 = -cosec(Ï€ / 4) = cosec(- Ï€ / 4)

We know that the range of the principal value branch of cosec

Therefore, principal value of cosec

tan x = 1 = tan Ï€ / 4

We know that the range of the principal value branch of tan

âˆ´ tan

Let cos

cos y = -1 / 2 = -cos Ï€ / 3 = cos (Ï€ - Ï€ / 3) = cos(2Ï€ / 3)

We know that the range of the principal value branch of cos

âˆ´ cos

Let sin

sin z = -1 / 2 = -sin Ï€ / 6 = sin(-Ï€ / 6)

We know that the range of the principal value branch of sin

âˆ´ sin

Now,

tan

= Ï€ / 4 + 2Ï€ / 3 - Ï€ / 6

= 3 x Ï€ + 4 x (2Ï€) - 2(Ï€) / 12

= 3Ï€ + 8Ï€ - 2Ï€ / 12

= 9Ï€ / 12

= 9Ï€ / 12

= 3Ï€ / 4

We know that the range of the principal value branch of cos

âˆ´ cos

Let sin

We know that the range of the principal value branch of sin

âˆ´ sin

cos

We know that the range of the principal value branch of sin

Therefore, -Ï€ / 2 â‰¤ y â‰¤ Ï€ / 2.

tan x = âˆš3 = tan Ï€ / 3

We know that the range of the principal value branch of tan

âˆ´ tan

Let sec

sec y = -2 = -sec Ï€ / 3 = sec(Ï€ - Ï€ / 3) = sec(2Ï€ / 3)

We know that the range of the principal value branch of sec

âˆ´ sec

Now,

tan

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