Chord Maximisation

Question

I am currently going back through all the "Challenge" questions in preparation for exams, and for this I do not know where or how to start, any hints would be appreciated. Now the way I have approached it is to have turned it into a triangle ABC with the angle(BAC)=90 and angle(BCA)=$θ$. Using trig laws, I found if we call AC=$x$ then the vertical height BA=$x\tan \left(θ\right)$, so if the length of AC and BA combined lets say were called $L$ then ;
$L$=$x+x\tan \left(θ\right)=x\left(1+\tan \left(θ\right)\right)$
But when trying to maximise, I am having trouble, as $x$ depends on $θ$ and vice versa, and I can't seem to be able to find the relationship between the 2.

Answer 1

Let the centre of the circle be $O$, and let $\angle BOA=\theta$. Then our sum $AB+AC$ is $1+\cos\theta+\sin\theta$.

It is enough to maximize $\cos\theta+\sin\theta$, or equivalently to maximize $(\cos\theta +\sin\theta)^2$, that is, $1+2\sin\theta\cos\theta$, which is $1+\sin(2\theta)$. This reaches a maximum when $\theta=\dots$.