 Hello and welcome to the session. In this session we discuss the following question which says what is the principal value of sin inverse of minus root 3 by 2. Before moving on to the solution, let's discuss the definition of the principal value of an inverse trigonometric function the value of an inverse trigonometric function which lies in the range of principal branch is called the principal value inverse trigonometric function This is the key idea that we use for this question Let's proceed with the solution now We are supposed to find the principal value of sin inverse of minus root 3 by 2 So for this we suppose that y be equal to sin inverse of minus root 3 by 2 So this means sin y is equal to minus root 3 by 2 or you can say sin y is equal to minus sin pi by 3 Since we know that sin pi by 3 is root 3 by 2 or you can say sin y is equal to sin of minus pi by 3 by 3 is equal to root 3 by 2 The principal value is the close interval minus pi by 2 by 3 belongs to the close interval minus pi by 2 by 2 and sin of minus pi by 3 is equal to minus root 3 by 2 Since we have taken y equal to sin inverse of minus root 3 by 2 this root 3 by 2 has sin of minus pi by 3 So this could be equal to minus pi by 3 that is sin inverse of minus root 3 by 2 is equal to minus pi by 3 and this value belongs to the range of the principal value branch of sin inverse As we know from the key idea that the value of the inverse trigonometric function which lies in the range of the principal branch is the principal value of that inverse trigonometric function So therefore we can say that the principal value minus root 3 by 2 is equal to minus pi by 3 is our final answer This completes the fashion, hope you have understood the solution of this question