A F.5 student, Yan, discussed with me something about “complex angles" in trigonometric formulae, in lunch hour, he asked me the way of solving
I just post it here. Starting with the Euler’s formula
We can extend the angle to complex number by using the above formula. Replace by , we have
Now, putting , we have
By quadratic formula,
Here, we put as a complex number and try to find out values of and . Also, by putting into the Euler’s formula, we have . Continue,
, taking logarithm, yields
Last time, I’d derived the formula of to Yan, hence it is done.