 Hi and welcome to the session. Let us discuss the following question. It says find sin x by 2 cos x by 2 and tan x by 2 in each of the following. Now in this question we are given the value of sin x as 1 by 4 and it's also given that x lies in the second quadrant. Let us proceed on with our solution. Since in the question we are given the value of sin x, first of all we will be finding out the value of cos x and it will be 1 minus sin square x in square root. 1 minus 1 by 4 square root gives us 1 minus 1 by 16, 16 minus 1 by 16 and that is root 15 by 4. Since it's given to us that cos x lies in quadrant 2 therefore its value will be negative. Right? So we have the value of cos x as minus root 15 by 4. Now we can easily find out sin x by 2 cos x by 2 and tan x by 2. I hope you remember the formulas for sin it will be 1 minus cos x by 2. For cos x by 2 it would be 1 plus cos x by 2 and for tan x by 2 we use the formula sin x by 2 divided by cos x by 2. So that means we need to find these two values first. Let us find out on substituting it will become positive root 15 by 4 the whole divided by 2. Now this will give us the value as 4 plus root 15 by 2 that is under the square root. After rationalizing the denominator we get the answer as 8 plus 2 root 15 that is under square root divided by 4 that is not in the square root. Similarly we have the value of cos x by after all these calculations and rationalizing it will be under root 8 minus 2 root 5 15 divided by 4. Now we have here 8 plus root 15 in square root divided by 4 the whole divided by square root 8 plus 2 root 15 by 4. These two will get cancelled out and we can take out root 2 common from the numerator and denominator and here we had minus sin we are left with 4 plus root 15 divided by 4 minus root 15. On rationalizing after rationalizing that is we have the value of tan x by 2 as 4 plus root 15. So this ends the question that was given to us. I hope you enjoyed this and do remember how to rationalize the denominators for now.