2014 Paper 1 Question 12

(b)

$\tanh z = -i$

$\begin{aligned} \Rightarrow \frac{\sinh z}{\cosh z} &= -i \\ \Rightarrow \frac{e^z-e^{-z}}{e^z+e^{-z}} &= -i \\ \Rightarrow e^{2z}-1 &= -i \, ( e^{2z}+1 ) \\ \Rightarrow (1+i)e^{2z} &= 1 - i \\ \Rightarrow e^{2z} &= \frac{1-i}{1+i} \\ \Rightarrow e^{2z} &= \frac{1}{2}(1-2i-1) \\ \Rightarrow e^{2z} &= -i \\ \Rightarrow e^{2z} &= e^{-i\left( \frac{\pi}{2} +2k\pi \right) }, \quad k \in \mathbb{Z} \\ \Rightarrow 2z &= -i\left( \frac{\pi}{2} +2k\pi \right) \\ \Rightarrow z &= -i\left( \frac{\pi}{4} +k\pi \right), \quad k \in \mathbb{Z} \\ \end{aligned}$

NST First Year Maths

Some unofficial solutions to Cambridge NST IA maths papers

2014 Paper 1 Question 12

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