An exact tangent value from a rhombus

An exact tangent value from a rhombus

IB Year 6 | Grade 12
6 marks

The complex numbers \(a\) and \(b\) are given by \(\dfrac{2+2\mathrm{i}}{1-\mathrm{i}}\) and \(\dfrac{2\sqrt{2}}{1+\mathrm{i}}\) respectively.

Find the moduli and principal arguments of \(a\) and \(b\).[3]

In an Argand diagram, the points \(A\), \(B\) and \(C\) represent \(a\), \(b\) and \(a+b\) respectively, and \(O\) is the origin. By considering the quadrilateral \(OACB\) and the argument of \(a+b\), show that \(\tan\left(\dfrac{\pi}{8}\right)=\sqrt{2}-1\).[3]

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Answer:\(|a|=2,\ \arg a=\dfrac{\pi}{2},\ |b|=2,\ \arg b=-\dfrac{\pi}{4}\). \(OACB\) is a rhombus and \(\arg(a+b)=\dfrac{\pi}{8}\), giving \(\tan\left(\dfrac{\pi}{8}\right)=\sqrt{2}-1\).

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