An exact tangent value from a complex quotient

An exact tangent value from a complex quotient

IB Year 6 | Grade 12
4 marks

Express \(\dfrac{\sqrt{3}+\mathrm{i}}{1-\mathrm{i}}\) in the form \(x+y\mathrm{i}\), where \(x\) and \(y\) are real.[2]

Hence show that \(\tan\left(\dfrac{5\pi}{12}\right)=2+\sqrt{3}\).[2]

Solution:

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Answer:\(\dfrac{\sqrt{3}-1}{2}+\dfrac{\sqrt{3}+1}{2}\mathrm{i}\). Hence \(\tan\left(\dfrac{5\pi}{12}\right)=2+\sqrt{3}\).

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