Using Euler's formula

Algebra Level 3

$\ln(\frac{\sqrt{3}}{2}+\frac{1}{2}i) \times \ln(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i) \times \ln(\frac{1}{2}+\frac{\sqrt{3}}{2}i) \times \ln(i) = \large \frac{\pi^{4}}{k^{2}}$

Find the value of $|k|$ .

(The radian range: $0 \leq x \leq \pi$ )

Hints: $e^{i(\pi)} = \cos(\pi)+i\sin(\pi) = -1$ $\Rightarrow$ $\ln(-1)=i\pi$

Reference : Euler's Formula

The answer is 12.

1 solution

Tommy Li
Jun 9, 2016

$e^{i\frac{\pi}{6}} = \cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}) = (\frac{\sqrt{3}}{2}+\frac{1}{2}i)$

$e^{i\frac{\pi}{4}} = \cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4}) = (\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)$

$e^{i\frac{\pi}{3}} = \cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}) = (\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

$e^{i\frac{\pi}{2}} = \cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}) = i$

$\ln(\frac{\sqrt{3}}{2}+\frac{1}{2}i) \times \ln(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i) \times \ln(\frac{1}{2}+\frac{\sqrt{3}}{2}i) \times \ln(i) = i\frac{\pi}{6} \times i\frac{\pi}{4} \times i\frac{\pi}{3} \times i\frac{\pi}{2} = i^{4}\frac{\pi^{4}}{144}=\frac{\pi^{4}}{12^{2}}$

$k = 12$

Using Euler's formula

The answer is 12.

1 solution

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