The complex number \(w\) is such that \(w^2+ww^*=6+2\sqrt{3}\,\mathrm{i}\). Find exactly the possible values of \(w\), giving your answer in the form \(x+\mathrm{i}y\) where \(x\) and \(y\) are real numbers.[3]
The complex numbers found in part (a) are represented by points \(A\) and \(D\) in an Argand diagram as shown below. Points \(B\) and \(C\) represent complex numbers \(b\) and \(c\) respectively.
Given that \(ABCD\) is a rectangle with \(AD=2AB\), find complex numbers \(b\) and \(c\), giving your answers in the form \(u+\mathrm{i}v\), where \(u\) and \(v\) are real numbers.[3]
By finding \(\arg(b)\), find the exact value of \(\tan\dfrac{5\pi}{12}\).[2]
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