 Hello and welcome to the session. In this session we discuss the following question which says using principal value evaluate the following. Sign inverse of find 3 pi by 5. Let's proceed with the solution now. We need to find the value for sign inverse of sign 3 pi by 5 using the principal value. We know that sign inverse of sign x is equal to x. Therefore, sign inverse of sign 3 pi by 5 would be equal to 3 pi by 5. Now the principal branch of sign inverse x is the closed interval minus pi by 2 pi by 2 and this 3 pi by 5 does not belong to closed interval minus pi by 2 pi by 2. Now sign 3 pi by 5 can be written as sign of 3 pi by 5 which is equal to sign of 2 pi by 5 and 2 pi by 5 belongs to the closed interval minus pi by 2 pi by 2. Therefore, sign inverse of sign 3 pi by 5 is equal to sign inverse of sign 2 pi by 5 since here we have sign 3 pi by 5 is equal to sign 2 pi by 5 and we already know that sign inverse of sign x is equal to x. So, here it would be sign inverse of sign 2 pi by 5 is equal to 2 pi by 5. Thus, sign inverse of sign 3 pi by 5 is equal to 2 pi by 5. So, thus, this is the required value. This completes the session. Hope you have understood the solution of this question.