 Hi and welcome to the session. I am Deepika here. Let's discuss a question find the principal value of sine inverse minus 1 by 2. Now we know that if the function is y is equal to sine inverse x its domain is close interval minus 1 to 1 and the range of its principal value branch is is minus pi by 2 close interval minus pi by 2 to pi by 2 So this is a key idea behind our question We will take the help of this key idea to solve our question. So let's start the solution let y is equal to sine inverse minus 1 by 2 or let Sine inverse minus 1 by 2 is equal to y then Sine y is equal to minus 1 by 2 and this is equal to minus sine pi by 6 and This is again equal to Sine of minus pi by 6 Because sine of minus theta is equal to minus sine theta now the range of the principal value branch of sine inverse Close interval minus pi by 2 to pi by 2 of minus pi by 6 Is equal to minus 1 by 2 therefore principal value of Sine inverse minus 1 by 2 is Minus pi by 6 so the answer for the above question is Minus pi by 6. I hope the question is clear to you. Why and take care