this is equal to one-plus cosine one-plus co-signed of two theta all of that over two and my goal here is that i really just wanna get everything, well i just wanna simplify and maybe we will do calculus value of the numerator solicit cosine squared theta is equal to this so four times this is just going to be so four times this before divide by two instances would be two times this numerator so it's going to be too plus too co-signed of to data that's this term right here and then this term this term right over here we could use the tree guided me that signed of to data is equal to two sign of state a co-sign if data or you divide both sides right to you get a one-half sign of tuesday dot is equal to sign of feder co-sign of theta so this is this right over here this part right over here is going to be one-half signed up to date a bore multiplying it by three just going to be plus three halves sign of two data sighing of too glitzy thus part over here clearly simplifies this is the recent history right this is one over three plus to co-signed of to data plus thirty halves sign of to data now we really just looking for in the minimum we're looking for the minimum value of the denominator which would give us the same as the maximum value of the new murder or just be one over at this minimum values alexia how how lol we can get assuming were above zero t how low we can get for this denominator write your words will look for its minimum value so one day we can do just to simplify things the middle value of this is going to be the same demand the bid of this thing over right there to kind of confuses the problem the minimum value the replies to cosign to stayed up plus three have sign of two data is going to be the same thing is the minimum value of three plus on the screen to the substitution the tuesday it is equal to exodus simplify things a little bit you don't have to do that so three plus two co-signed of acts plus three halves sayed of acts so this is a pretty simple expression to see how we can figure out some minimum value and my temptation is to take the derivative find out where the derivative is equal to zero and then that will either be a maximum or a minimum point so let's take the derivatives the derivative of this expression right over here with respect to acs would really love three with the for texas zeroed route of two cosine of axis negative negative to sign of x derivative of three have side of actions will be plus three halves cosine of x and that is going to be equal to zero we want to find where the slope is your committee worry about somewhere minimum point and let's see we can admit we had to sign of extra boats side so we get the rehabs co-signed of acs is equal to two signed of x and then we can divide both the sides both sides of this equation by will survive like to first alaska too many steps of three forts co-signed of x is equal to sign of acs and to divide both sides by cosine of x so we get three over for is equal to sign of x over a co-signer of x which is the same thing as the pentagon that cs so uh... annexed by you that gives us three or attended of annexed by his three-fourths is going to give us either a maximum or a minimum point celeste think about this