Determine whether cosec - 1 3 2 is defined or undefined, if defined then find the set of values.

Given: cosec - 1 3 2 Domain of cosec - 1 is R - ( - 1 , 1 ) i.e, ( - ∞ , - 1 ) ∪ ( 1 , ∞ ) . Here, the value 3 2 is not within the domain. Therefore, value of cosec - 1 3 2 is undefined.

For the principal value, evaluate the following: sin - 1 3 2 + cosec - 1 - 2 3

Given that, sin - 1 3 2 + cosec - 1 - 2 3 = sin - 1 3 2 + cosec - 1 - 2 3 - = sin - 1 sin π 3 + cosec - 1 cosec - π 3 = π 3 - π 3 = 0

For the principal value, evaluate the following: sec - 1 ( 2 ) + 2 cosec - 1 ( - 2 )

Given that, sec - 1 ( 2 ) + 2 cosec - 1 ( - 2 ) = sec - 1 sec π 4 - 2 cosec - 1 cosec π 4 = π 4 - 2 × π 4 = π 4 - π 2 = - π 4

Find the principal value of the following: cot - 1 ( - 3 )

Let cot - 1 ( - 3 ) = y cot y = - 3 We know that the range of the principal value branch is ( 0 , π ) cot y = - 3 = cot 5 π 6 ⇒ y = 5 π 6 ∈ ( 0 , π ) Hence, the principal value of cot - 1 ( - 3 ) is 5 π 6

Find the principal value of the following: cot - 1 ( 3 )

Let cot - 1 ( 3 ) = y cot y = 3 We know that the range of the principal value branch is ( 0 , π ) . cot y = 3 = cot π 6 ⇒ y = π 6 ∈ ( 0 , π ) Hence, the principal value of cot - 1 ( 3 ) is π 6 .